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Same base as in my question about feeding an underground base (in Abbeville, SC).
As you go underground, the air heats up. What would be a reasonable depth so that (a) it could be easily cooled,(b) it must not be noticiable to the outside world, and (c) it could support hydroponics?
Heat underground is not uniform. If you have an underground base in a geothermal active area, you're probably not getting very deep before things heat up real fast. On the contrary, if you are away from geothermal activity, and close to water (underground river, or simply under/next to a lake or sea), then you'll be ok.
Edge case to prove the point—Mariana Trench is 6.831 miles below sea level at its deepest. Here it's 1-4°C (pretty cold), so if your base was near this trench, the problem would instead be heating!
Best case—Under a lake. You can have tunnels out, or you could have an entrance through the lake itself. You're not in danger of heating, you've got a new food source, and you've got your water/sewage problems pretty much solved (although make sure you source your water far away from your sewage!)
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.
Heat Sinks
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)
Conclusion
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.
Use water cooling and interchange your waste heat to the water table. Side-drill pipes as long as you need to keep from ingesting your discharge water and to move the heat signature (what little of it there'll be) away from your base.
If you discharge your heat in a large enough area, it will be inconsequential and will take a sharp eye to spot (why is this land a half degree hotter than surrounding areas); this is how the Soviets found the Red October in Clancy's book. You may be able to mask even that by well chosen discharge sites.
