In your words, why is a cool basement often damp in the summer

I believe it's because most basements are not insulated casing it to be cool and due to rain that is what causes it to become damp.

At the time of summer season, the basement would be comparatively cooler in comparison to the external temperature. Also, in summers, the basements get warm as well as humid. When this humidity enters the basement, the moisture in the air gets condenses into the cool surfaces.

This leads to a typical dampness of the basement, and this dampness can often turn into some kind of mustiness, as various basements exhibit poor aeration to circulate fresh air through them.