Unless your base is out in the frozen tundra, cooling is only done by cycling air (same as air conditioning in any other large building around). And this will not so much be affected by depth as by the size of your base, the equipment you're running, the number of people running around, etc.
The deeper you go, the hotter it gets, so there's no escaping that. You will need powerful systems to push air out, as well as pull it in (also filter it, etc.) Your vents can be masterfully hidden in caves, etc.
However, you can decrease your reliance on this system by:
Recycle Your Air
There are various ways in which you can scrub CO2 out of air. You could feed it into your greenhouse, etc.
Water is one of the best heat sinks there is.
If you encounter underground rivers or springs when you dig down you can reroute the water and use it to run a cooling system, so that you don't need to dump that heat out, and risk exposing your position. (the opposite - heating - is easy: geothermal, or nuclear)
Depth does not affect hydroponics implementation, and cooling is going to be dependent on either running into underground bodies of water, or your ability to run large amounts of air through your facility, and back up to/from the surface.