If in the future the human race spreads out in the solar system I would think an outpost orbiting in the vicinity of the heliopause (the furthest extent of the solar wind's influence) might be reasonable. What would be the next reasonable staging area away from the sun's gravity well?
At first blush, there doesn't seem to be much point to an outpost at the heliopause, at least not from the perspective of an interstellar voyage.
The heliopause is about 123 AU, or 0.0019 light-years. On the other hand, the closest other star is Proxima Centauri at 4.24 light-years, or 2230 times the distance. Even in the best case, it seems hard to believe that resupplying for the first (or last) 25th of a percent of the trip is worth the extra difficulty of hitting the fuel depot. Efficient launch windows to Mars are only open for a few months out of the year, and it's practically our next-door neighbor in comparison. I don't even want to think about how rare your ideal launch window would be for a heliopause outpost.
The other problem, of course, is getting the supplies to the depot. In theory, a station on Mars, one of the gas giants' moons, or even a TNO could use resources present there to create fuel for passing ships. But in the heliopause, there's nothing to harvest. Any fuel we load onto ships there will have to be laboriously transported to the station from elsewhere in the solar system, cutting drastically into the overall efficiency of our scheme. And the station will itself require power, and life support if it has crew.
Ultimately the added complexity of building and maintaining the station itself, stocking it with supplies for ships, arranging for pickup of those supplies, and coordinating launches around the orbit of the outpost (or worse yet, two outposts, one in each system) would simply be impractical compared to the expensive, yet very simple approach of increasing the ship's fuel and supply capacity.
One of the big problems with having an "outpost" anywhere in space is you need to expend considerable energy and/or reaction mass to match speeds with it. If you were on your way to someplace like Alpha Centarui, slowing down to meet an outpost and then speeding up again to resume your journey would essentially negate any sort of advantage that such an outpost might offer.
Indeed, if you are using beamed energy, such as giant lasers orbiting close to the Sun to push laser light sails, you would have done your acceleration deep inside the Solar System and already be blasting past the Heliopause at a considerble velocity. Slowing down and stopping isn't an option here.
What may be a useful option is to mount a giant laser and reflectors in the Oort cloud to provide two things: shining the beam ahead of the oncoming spaceship to illuminate and hopefully "push" aside dust particles and gasses that might impact the ship as it passes through interstellar space (as you start moving at higher speeds, these particles become dangerous. Knowing where they are to steer around them or pushing them away makes the ship's journey safer).
The other advantage would be to establish a ""photonic railway". Using the laser to illuminate a light sail provides extra thrust, and having a series of reflectors allows the light to be "recycled", bouncing between the lightsail and the reflector and them back to the lightsail. Having a "station" in deep space allows you to accelerate the spacecraft again as it passes, allowing for higher transit speeds while also freeing the solar laser to power another spacecraft. This deep in space you would likley be mining comets for deuterium in order to power fusion reactors to drive the laser, but since the material is already there and available, the overall cost is failry low.
So the idea of an "outpost" needs to be reconsidered somewhat in order to make it useful for interstellar flight.
In the immortal words of Douglas Adams, "Space is big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to space."
Because it's so big (as in, billions of outposts around the heliosphere wouldn't stop a Romulan invasion), you actually need a good reason to have an outpost. You didn't propose one, so I'm going to somewhat ignore your question and answer the question, "What reasons would require a human outpost at our solar system's heliopause?" Hopefully, that will answer your question.
Good Reason: There was a day long before GPS when giant concrete arrows could be found all over the United States. Their purpose was to help guide early airmail pilots. Now, you may think that keeping that big bright sun in the crosshairs is easy, but 3D navigation is harder than you might think! So outposts serving as transponders would help navigation. Six would be all you need, broadcasting in all directions. Your ship need only pick up three of them (more would be better) to triangulate the approach to Earth — especially if Earth had rules governing exactly how an approach could be made.
Poor Reason: Refueling. Honestly, the size an outpost would have to be to deal with resupplying incoming or outgoing ships is planet sized. As in Death Star sized. You'd have to have something large enough to grow its own food and hope to never need another piece of metal ever. The cost of resupplying the outpost is well beyond its value. Just build bigger ships.
Kinda Good Reason: Such outposts could act as a detection net. If you had enough outposts (possibly millions or billions), you could use light and, when someone was detected, forward that data to Earth. The incoming ship must slow down (unless it's a strafing run) so the light will always arrive first. Or you can use a trope (tachyons, etc.) to achieve faster-than-light detection.
Good Reason: (Thanks @David258!) Scientific exploration. Similar to the Hubble telescope being outside our atmosphere being a good thing, a research station outside the heliopause would allow for incredibly clear views across the electromagnetic spectrum.
Good Reason: (Thanks @JoeBloggs!) Diplomacy and customs port of call. This would inevitably become a trade station (or at least an aggregator to reduce traffic around Earth itself).
And that's all I've got.
As for more distant outposts, the only reason I can think of would be trading posts at cross roads. In this case you'd want a big ol' death star sized object at or near where multiple trade routes between multiple stars cross. While this would make a pretty good scifi story stage, in reality it's probably so unlikely for trade routes to cross close enough to each other (see Douglas' quote) to justify the outpost.
One thing to keep in mind, though, what we just learned when Voyager passed the heliopause is that intersteller space is HOT. As in 89,000℉ hot. That's hotter than our sun's photosphere. Interstellar space travel has some huge problems to overcome. But, to be fair, if you have the tech to travel, you have the tech to outpost.
Astronomers Believe Jupiter Kicked a 5th Gas Giant Out of the Solar System 4 Billion Years Ago. It Might Be Hanging Out In Or Near the Heliopause
The Nice model of solar system formation proposes the existence of a fifth gas giant planet during the initial formation of the solar system, but was later ejected as either a rogue planet, or sent into a very deep orbit by a close encounter with Jupiter.
If planet Nice exists, even if it is wandering but not yet left the Oort cloud (which is almost a light-year thick), it might possess moons (it's a Neptune-sized planet) and other resources that make it a natural waypoint for anything happening at the extreme edge of the solar system.
Research. It may also be a good place to study interstellar space with some safety: humans can live behind the protective wall of the Neptune-sized planet's magnetic field, but interstellar space is a very short rocket-ride away. It might make a lot of sense to spend a few years getting very smart about engineering and crew health in interstellar space before we commissioned a first interstellar vehicle that might spend decades in the stuff.
Gravitational assist. A ship launching from the inner system might, instead, use Nice as a final gravity-assist before hitting the long leg of a journey between stars.
Emergency alternate. Even if Nice isn't on the course, it might be planned to pre-settle Nice and route the long voyage near Nice's orbit. If anything breaks in the first several outbound months, Nice could be an alternate destination for that emergency.
the oort cloud is a region of the solar system that is beyond the planets, and would be humanity's best bet for an outpost because it also happens to be an extraordinarily great hiding place.
I expect to visit other planets, and even other stars and galaxies. The first trip will be to Yuggoth, the nearest world fully peopled by the beings. It is a strange dark orb at the very rim of our solar system... “There are mighty cities on Yuggoth—great tiers of terraced towers built of black stone like the specimen I tried to send you. That came from Yuggoth. The sun shines there no brighter than a star, but the beings need no light. They have other, subtler senses, and put no windows in their great houses and temples. Light even hurts and hampers and confuses them, for it does not exist at all in the black cosmos outside time and space where they came from originally. To visit Yuggoth would drive any weak man mad—yet I am going there. The black rivers of pitch that flow under those mysterious Cyclopean bridges—things built by some elder race extinct and forgotten before the things came to Yuggoth from the ultimate voids...
Yuggoth is also a fine staging area for further exploration.
But Yuggoth, of course, is only the stepping-stone. The main body of the beings inhabits strangely organised abysses wholly beyond the utmost reach of any human imagination. The space-time globule which we recognise as the totality of all cosmic entity is only an atom in the genuine infinity which is theirs.
Yuggoth is well sited for your outer base, and has considerable building stock which can be retrofitted for your uses. The locals can advise as as to which pits are occupied and should be avoided, and can also help your explorers access the genuine infinity which is theirs.
After reading the Trilogy of the Three Body Problem, I think that the Asteroid Belt, Jupiter and Saturn would be good places to have an outpost.
If you want to sent a spaceship to another star, Jupiter is nearly a must, at least for the gravitational assists. And if you have an outpost there, even better because you will be able to send an unmanned resupplier without needing to decelerate the spaceship.
That is supposing that we stay with chemical propulsion.
Outpost? For what? To supply interstellar ships? But a "ship" is a lousy means of interstellar travel.
You don't make a "ship". You make a nearly self-sustaining space habitat and then move it. You colonize a medium-sized Kuiper Belt object, tens of km radius. That has all the raw material you need for anything, including plenty of deuterium for your fusion power plants, maybe enough for a few million years. All accessible by tunneling through a low-gravity body. Humans live in big wheels, several km in radius, spinning for healthy gravity.
At some point, you decide that the solar system isn't where you want your descendants to be, so you spend some of your mass via mass drivers or plasma thrusters and leave at modest velocity. 250,000 years to Fomalhaut at 30 km/s. Lots more small bodies with easily extracted resources there.