Two years ago I had my basement renovated and the contractor installed an air conditioner in the wall that sticks out just above the ground level outside in the backyard. When I run my air conditioners in the other parts of the house I notice condensation that drips when they run a long time. I don't see any runoff in my basement air conditioner and I'm wondering if I should be concerned with moisture running inside my walls into the foundation? The wall on the side the air conditioner is cinderblock that they sheetrocked over and painted.

Yes you should... the condensation that forms has to be going someplace... better to find out where sooner rather than later.