The answers so far are awesome. They talk about energy/power, water, air quality, aesthetics, and warfare. There is a line about "transportation". Mining. The assertion that the waterfalls were there and they did what they had to.
But I could add a few others ....
Your pictures show overshoot waterfalls, which allow a reservoir behind the falls that is higher than the water in front of the falls. Given the Gibbs free energy surface of water, that is to say, given that water bonds have high energy, and nature goes to the lowest energy state, there is a preferential movement of stuff in the water to the surface. Alcohol does this. It could be a mechanism for separating things that are somewhat soluble (think stills and moonshine), but also have a lower Gibbs free energy surface, like an open-air refinery and storage.
Ponds like that could also be about sky-cooling. (It is one of my favorite techniques, and a technology lost to the modern world). This guy is selling silicon, so he contrives his answers only in terms of silicon. Silicon is not required. He asked the genii of his mind a weaker question, and received a weaker solution.
Water isn't always liquid. It freezes. It evaporates. What happens when the freezing cold of winter hits those styles of falls. At least half the volume is stored as liquid behind the (very insulating) ice of the frozen fall. It could be about retaining liquid, possibly retaining biodiversity such as aquaculture fish, or magical protozoa, or such. Humid air is heavier than normal air, so the pit-like region could be about retaining humid air.
There could be a half-and-half where the "humid" "air" forms mists, or micro-clouds, that have a purpose.
Water is made of hydrogen and oxygen. Perhaps it feeds an oxygen converter? Fusion power-plant? Deuterium-oxide (heavy-water) or tritium-oxide (very heavy water) mining for nuclear-energy uses?
It could be used to transport oxygen. Modern fish-tanks have waterfalls that entrain and dissolve oxygen so the fish can live. There are other gasses that could be entrained for life. The available gasses depend on the atmosphere and the nature of the life in the water.
The blocks at the lower part of Logans run are for turbulating (and slowing down) rushing floodwaters. Rivers don't always run at the same height. When there is rain upstream they run high. Perhaps this is about enduring the larger climate instead of the local comfort. Perhaps, like the cathedral under Tokyo (link), they help the city to endure natural disasters.
I like to think about things like chemically pumped lasers. Ever hear of the Jello-laser? What if it is carrying particulates of metastable lasant through a beam-line? You would need something about the waterfall to not poison the metastable state, and you would need a long enough metastable lifetime. The "COIL", chemical oxygen iodine laser, had long metastable lifetimes. I often wondered if it was a dual-use technology, because the iodine could be used in the 3rd stage of fission-fusion-fission nuclear pumped lasers as part of a 3-stage nuclear weapon, or even a star-wars-esque nuclear pumped super-laser for engaging a ground-based target with mega-scale laser irradiation. If there was a material with a long-enough metastable lifetime, the civilization could be filtering it from the water, from a far past war, filtering it from the water to power their infrastructure, or rebuild the terrible ancient weapons.
I thought about this for a long time, but I think there may be waves in the oceans that have been moving since the big bang, that carry some of that information. I have also thought that if there were two places that had a "structural weakness" in the universe where it would be easier/easiest to make a worm-hole, and one of them was at an ocean, the waves would show it. An ocean that is a billion years old, could have a billion years for the gravitational or inertial perturbations of the wormhole-end collected in a way that is encoded in the waves. I have thought about the Nikola-Tesla + Philadelphia experiment, and wondered if Tesla's inverse free electron laser, meant to absorb radar for radar-scale invisibility, by oscillating the ship at the frequency of the radars, stumbled onto such a phenomena. By inverted, I think he turned a FEL inside out, so the beam became like a surface on the outside of the ship. If an ocean had those Tesla-Philadelphia waves, then a super-intelligent race might use a waterfall in the ocean to try and distill primal oscillations in order to build doorways to other worlds, or other dimensions, or to measure far-away parts of the universe remotely.
It could be about plumbing. Dinosaurs make a big poops.
I've seen pneumatic logic circuits. It could be a hydraulic variation, where it is performing logic or computations. If it were the right material, it could be a quantum computation. See slide 41 of this to show macro-scale quantum-mechanics (top row of table) connects Heisenberg scale to Kolmogorov scale, its on the order of 10 microns, or about 1% of a millimeter, but who is to say that a large-scale structure running a surface of such quantum computations would not look like a waterfall.
In a world at war with marine creatures, they could be used through giant pipes to make very loud, very specific wavelength, underwater sounds, like active sonar, which devastates current cetaceans. It would be like a giant pipe-organ, for water, that makes sounds that fend off war-parties. You could make sounds that are loud above ground too.
Could they power hydraulic and/or steam cannons? Water powered resets of siege warfare equipment. Can we make a water-driven reset of a trebuchet? We could make ice rocks to shoot, or ice arrows to shoot. It could provide both ammunition and motive power.
I lived in the Sonoran desert for 16 years (Phoenix), and one thing that happens every time it rains is desert spores reproduce like mad. I had a friend who was allergic, and when they lived downwind of a football field, they had alergy attacks every time the field was watered. The waterfall could be about aerosolizing water, to trigger molds or other organisms.
Update (Jan 2019):
- They could act like a giant "bong" in filtering part of an aerosolized "medication" so that the remainder was a higher quality in some way.
- If the civilization had a "infinite cold source" then it could be used to turn liquid water into freezing clouds of ice. I've thought about it some, but there are going to be optimal crystal configurations for ice-replication. There have been sci-fi stores about building ice-format that self-replicate and "destroy the world" into an ice-ball. If they exist, those are encoded at a microscopic scale in the axes and shapes of snowflakes. I think you could make micro-snowflakes with great Stokes numbers (so they stay in the air for a long time) with surface roughness/smoothness such that they are very effective reflectors of light, so they change albedo. I could see something like this as a doomsday weapon against a world, to turn it into ice by injecting enough of those crystals, at a high enough rate, to destroy the world, and make it fight against recovery - to make it resist ability to sustain life forever. It might take simple oscillators and flow directors up near the top, and the right kind of blower a little farther down. I've wondered if China/Mongolia could make something like this that injects into the jet-stream, and turns Alaska into an iceball while making a solid ice-bridge at the Bering strait. How long does that area have to be kept at 40-below before the ice gets "deep enough"? A counter-waterfall could "drown out" or have chiral anti-snowflakes that either reduce below critical concentration, or selectively target and capture the problem flakes during interactions. This would allow two competing massive waterfall/snowmakers to fight it out, where one sustains a warmer climate, and the other tries for ice-ball.
- If they wanted to a civilization with giant acoustics, and existing clouds, might want to try and induce the super-thermal-radiant surface topology into the top of the cloud layers, so they might make very particular acoustics to make non-snowflake based ice-doomsday weapons against the world.
- Converting gravitational potential energy into heat for some warmth increase. (link) In a high gravity world, the potential heating effects are much higher. There is an interesting effect that the center of the planet is weightless. There is a linear decrease in "weight" as you go toward center. If you wanted to stake a "Krypton" energy source, then redirect the gravity waves, and put huge but arguably sustainable crust stresses on the entire world in order to make gravitational potential energy into a "hotspot" for energy extraction. Gravitational wave teleportation? Can gravitons tunnel? Can they be induced into spatial (or temporal, because they are the same thing and the scanning-tunneling microscope is really time-travel too) tunneling?
- Machining. One could use it as a giant tumbler. If you did it right, you could concentrate the pressure (think about how waterjets make mach 3 water) and use it for cutting, for pressing, for bending, for "peening"
- I have been thinking about the meta-meaning behind myths, and how the deep past can strongly speak, in much more meaningful and powerful ways that science can, to the present and future. This is in the sense of Jordan Peterson, such as here. The waterfall could speak something of the deep past to the deep future in a meta-meaning sense. Fruit trees are important to human development, and so something like this could apply.
- If we think about a "living planet" like the organ equivalent of the difference between an endo-skeleton and an exo-skeleton such as grasshopper, then water carries nutrients and acts like hemolymph or blood. It could be argued that the air currents are a second type of "blood". If the water is blood for the inside-out massive meta-organism then all that blood is, the water is. Transport. Immune system. Where life of the cells comes from. In this paradigm, a waterfall is where the two different "breeds" of "blood" interact. For an organism, where two fluids interact, there are very specialized tissues, and there is immune activity to prevent propagation of disease. Not only can water transform the gas, but he gas can transform the water. In an organism substantially more advanced than an insect, the lymph nodes are where blood and lymph mix and interact - they are a critical part of the immune system. One could argue that rocks, for a living planet, are like bones. So the waterfall is air and water and stone - blood and lymph and bone.
- I like Brian Deragon's answer, but we could do a bit of both. What if there was flaming oil on top of water? Coal, when heated, can burn underwater (link). What if there was submerged burning substance as well as floating ones? Defense? Offense? Disposal of waste?
- When I look at the waterfall, I think the sea is lower. What if it isn't. There are dikes in the Netherlands, and when they leak it isn't a good thing. What if it isn't about using something that is going away to somewhere else (the water in the river) but is instead about something scary coming close? What if the dam is about the coming ocean, not the leaving river?
- Appeal to the unknown: when you (as a technical nerd) want to solve an otherwise impossible problem, you know the center of your domain isn't where it will work, so you go to the half-lit edges. You don't go to the darkness, but to the shadow-areas. It could be that this is the imagination-equivalent of a shadow area in that, psychologically speaking, the presence of these style of waterfalls accelerates suspension of disbelief toward futuristic fictional universes. It might be a generational or cultural thing, but it might be a universal psychological mechanism.
- I recently watched the pilot episode of "Dragon Pilot: Hisone and Masotan". When we think about cooling, there was a sense of "air-conditioning" but this could be a way to cool off a dragon. To cool some highly exothermic, semi-aquatic lifeforms.
- Salmon leap waterfalls to get to their breeding grounds. Those who don't make it, don't breed. It could act as a form of natural selection for salmon or other aquatic organisms. It is also a route for fresh baby salmon to get to the ocean, so instead of reduction it could be a protected transport corridor that typical in-ocean predators cannot overcome. It doesn't have to be for salmon, but could be for tiny things or for huge things.
EDIT Apr 2020:
In a world of telepaths, the wavelengths of electromagnetic radiation are about the diameter of the planet, and fluids like water can transform the paths of the emanations, so for telepaths a waterfall could act like "Cerebro" on X-Men (fantastic stories).