After patching the stone foundation in the basement with with regular drying mortar/cement, how do I get the cement to dry quickly if the basement is damp from humidity?
Cement (and gypsum) compounds don't need to dry to harden. Water is one of the hardening reaction components. This is why when the weather is dry or windy you actually need to add water to the concrete surface to prevent it from over-evaporating.
I don't remember the exact numbers but something like after 8 hours the cement compound has passed through the initial reaction phase and started hardening and at that point you can add whatever amounts of water - it simply won't matter. I remember in a book on repairing wells I read about the underwater repair procedure - you pump out the water so open the damaged spot, replace the bad bricks, render the surface and then you have to keep the mortar out of the water for just several hours (I don't remember the exact number, something like in between 2 and 8) and after that time you can allow water to flood the repaired area and that's not a problem for mortar in any way.
So the bottom line is - you're trying exactly the opposite of the right thing. You need to keep hardening cement compound wet, not dry, so that is hardens well.