How do I cool my base?

Base Cooling with Aquatuner and Steam Turbine
The Dangers of Heat
Once your colony is no longer in immediate danger of running out of oxygen or food the next major threat to your survival is heat. Usually the way heat kills your colony is by stifling your plants and thus causing your duplicants to starve rather than by destroying the base directly. Protecting your crops by insulating your farms can greatly increase how long you can go without cooling your base but eventually the heat will have to be dealt with. Heat can be somewhat mitigated using an Anti Entropy Thermo-Nullifer or wheezeworts but for a more robust solution I recommend either using a Thermo Aquatuner with a Steam Turbine or circulating the cold water from a water geyser.
Thermo Aquatuner and Steam Turbine
By far the most powerful solution for dealing with heat is to combine a Thermo Aquatuner with a Steam Turbine. Even just a single Thermo Aquatuner and Steam Turbine, commonly referred to as an ATST, can provide enough cooling for most reasonable colonies.
The first step to removing heat from your base is to gather the heat and move it to somewhere where it can be dealt with. To accomplish this we use a liquid pipe with either water or polluted water coolant that circulates through your colony. As the amount of heat that needs to be moved at any one time is generally low we can get away with using a normal pipe made of granite for most of the loop. Granite pipes won't transfer heat quickly but slow and steady is enough for most of the colony. For any problem areas with lots of heat we can switch to using radiant pipes made from refined metal which will exchange heat much more quickly.
Once the heat is in your cooling loop the next thing to do is to move it somewhere that it can be dealt with. To do this we use a Thermo Aquatuner made from steel in a room filled with steam. When provided power the Thermo Aquatuner will move the heat from the coolant into itself which will then be transferred to the steam. When selecting coolant for a Thermo Aquatuner it is important to pick one with the highest possible specific heat capacity for a liquid at our desired temperature range. In most instances the best choices for coolant are polluted water, water, nectar or if you are rich supercoolant.
In order to prevent over cooling and freezing the coolant in the pipes it is important to regulate the Thermo Aquatuner to only run when the coolant is above the desired temperature. To do this we use a Liquid Pipe Thermo Sensor directly before the input to the Thermo Aquatuner and we configure it to send a green signal when the coolant is above our desired temperature. For cooling a base I find that setting it to above 25℃ provides a comfortable temperature. We also need to keep the coolant circulating in the pipes when the Thermo Aquatuner is off and to accomplish this we use an arrangement of two bridges that provide an alternative path for the coolant.
The final piece of the puzzle is to use a Steam Turbine to delete the heat from the steam and convert it into power which refunds some but not all of the cost of running the Thermo Aquatuner. It takes steam from the room below, converts the heat into power, and then returns the steam in the form of water which we can send back into the steam room. The Steam Turbine itself must be kept under 100℃ so it must be cooled but luckily we have a pipe of coolant nearby so we can use that to cool it. Since the Steam Turbine needs to be kept well below the temperature where it would start taking overheat damage making it from steel provides no benefit so be sure to use whatever refined metal you consider cheapest.