A type 1 civilization is defined as a civilization that can utilize the maximum amount of energy equal available to its entire planet, roughly 1017 watts for Earth.
This is equal to the amount of energy that the Earth receives from the Sun. Assuming that the Type 1 civilization doesn't cover its home in solar panels (and instead got the 1017 watts from other energy sources, like fission/fusion), how would the civilization avoid boiling its own oceans and destroying its planet?
Comparing a civilization that consumes 10,000X more energy than we do planet-wide to our technology is comparing apples to coconuts
You (and some of your commenters) are assuming that by the time our civilization advances to Kadashev Type I status that we haven't improved the efficiency with which we consume power. That's a little short-sighted.
A good example of the problem is comparing a simple AM radio from the good old days (say, 1950s) to the transistor radios of my youth (1970s) to the iPod-ish stuff of the late 2000's.
You could literally cook an egg on those old radios (I've had the joy of repairing them. Sing it with me, children! We love radio tuuuuuubes!).
My transistor radio (emblazoned with a lovely red, white, and blue "76" denoting that banner year) fit in the palm of my hand (if not in my pocket) and ran on a 9v battery. By comparison it created no heat at all (although it did a bit).
And the iPod was basically a device an inch square and a quarter of an inch thick using a 1.5v lithium button battery that could play music forever compared to my transistor radio, clear music to boot, and its heat generation was basically nada, zilch, (almost) zero.
Efficiency is one of the hallmarks of technological advancement. The assumption that tomorrow's fusion reactor will generate as much heat-per-kilowatt compared to yesterday's Three Mile Island is ludicrous.
So, how do they keep from boiling their oceans and igniting their atmosphere?
By using power a whole lot more efficiently than we do. We can speculate that this will include things like room-temp superconducting materials, molecule-thin insulators on micron-sized metal windings for motors, much lower conducting resistance and much higher insulating resistance in our semiconductors, etc., but the reality is we have no blooming idea how this will happen (any more than we know how to practically generated 10,000X the energy our planet uses today). But, you didn't (and shouldn't) ask how...
Technological efficiency would increase with all other kinds of technology, allowing the consumption of ever greater amounts of power without burning up the planet.
A Type I civilization can utilize the amount of energy that's available to their planet, but that doesn't mean they're actually utilizing the specific energy that reaches their planet.
They could have moved their entire civilization out into artificial space stations fed by an array of solar receptors in orbit around their sun, while their home planet was cleaned up and reclaimed as an uninhabited nature preserve. As long as the amount of energy those space stations were able to utilize happened to add up to at least the same amount of energy that their home planet receives from their sun, they're a Type I.
There's two things you can do.
1) Don't make so much heat in the first place. Other answers have covered this pretty well, especially JBH's, so I won't get into it.
2) Make your civilization radiate that heat away faster. We already have metamaterials which radiate heat extremely effectively. Check out this Ted Talk about them. So what you do is you build your cooling systems out of these metamaterials. They radiate heat in specific infrared wavelengths such that as little as possible is absorbed by our atmosphere. Imagine the cooling pond of a nuclear reactor, but none of the heat stays in our atmosphere. Some of it will get absorbed by the ground though.
If you need to avoid heating the ground as well, you can use a Space Fountain, using your coolant as the supporting fluid. By letting the coolant cool off high in space, less of the heat will be radiated towards the ground. It may be possible to also use convection to "pump" the water up the fountain, so you don't even need additional pumps. Space fountains kill two birds with one stone, too, because a Type 1 civilization will likely want a lot of orbital infrastructure as well, and a space fountain makes a good launch platform if it's tall enough.
You get "cooking the oceans" from heat, more specifically waste heat. Not all uses of energy result in waste heat, not all uses of energy result in waste heat in places we care about, and then even if it does there are countermeasures.
We use antimatter generators to create any level of energy. We use Star Trek transporter technology to "create" matter itself. Then we connect the two with superconductors. It would take absurd levels of energy to create matter itself, but that matter isn't going to be heating the planet.
If you don't like Star Trek then that civilization could be using high levels of energy to push stuff around in space. At it's absurd extreme, pushing a planet from one orbit to another (or blowing one up), wouldn't affect another planet's temperature.
If you have serious problems with waste heat on the planet then there's geo-engineering, i.e. blocking out some of the sun, or even absurdly large energy pumps releasing/radiating heat/energy into space.
