We too run a basement dehumidifier in season. It protects everything, including my ammo and reloading components. Without the dehumidifier, everything in your basement will eventually smell of mold and mildew, especially anything stored within 18 inches of the floor.

I've used ammo stored in damp environments and it worked just fine. As for storing ammo in a "cool & dry" place, I think this was more applicable when we still had paper shot shells as they would swell with moisture.