Last yr went downstairs one day similar to this years heat and humidity combination and had a lot of condensation dripping from ductwork.  I had been in habit of opening basement windows in the summertime all my life, but after listening to Doug Rye, an expert on energy cconservation and house building, I got the idea to close all outside air entrances to the basement and it solved my condensation problem.

Outside, humid air was coming in and hitting the cool ductwork and causing the condensation.  Once the source of the humidity was shut off and all the air in the basement was "conditioned," my "rain" in the basement ended.