Question

why the cool basement is damp (high relative humidity) in the summer

Answer 1

It may not be that the basement is damp, but that your perception is that it is damper. With summer heat — depending on where you live — the humidity outside and in the upper floor(s) gets lower. This would lead to a contrast effect, making the basement seem damper as you enter it from above.

I hope this helps a little more. Thanks for asking.