Reducing sound transfer of finished basement drop ceiling on a budget
Sounds travels through vibration; one molecule bumping into another.  It travels through the air, but it also travels well through relatively thin solid material (ie, wood).  What you need to do is block those vibrations.

As you mentioned in your post already, the easiest ways to help your situation would be sound dampening insulation in between the floor joists, and/or sound dampening ceiling tiles.  Another useful step would be to decouple the frame of your drop ceiling from the floor above, by placing rubber grommets/mat between the drop ceiling hangers and the floor joists.