If it is a non-standard AC installation it might be tough for folks here to provide helpful comment,
But if it drips when it is NOT raining and drips much worse when it is raining, it sounds as though you could have two issues,
(1) AC design is usually such that normal condensate created on the evaporator from the AC operation is normally routed to a drip tray that has a connecting pipe running in the wall to drip out into the trailer wheel well outside the trailer - you should likely see a rubber tube hanging in the wheel well to let this water escape onto teh ground.
Possibly the drain line can be blocked and if so you might be able to clear it by gently blowing compressed air up into the drain pipe.
(2) If you get a lot of water coming in when it rains then you may have trouble with the foam sealing gasket used to seal the AC unit onto the roof. The gasket alone should provide all the seal you need against rain water entry - normally there should not be any caulking material.
I don't know your design, but in ours there are three or four bolts (forget which!) accessible from inside the trailer when you remove the inside AC grille, and it if so, slightly tightening these bolts to compress the gasket more might help.
Hope that is of some help.
Brian & Connie Mitchell
2005 Classic 30'
Hensley Arrow / Centramatics
2008 GMC Sierra SLT 2500HD,4x4,Crew Cab, Diesel, Leer cap.