When smoking in cold weather, I always have that problem. It is from condensation forming when the moist hot air escaping from the Bradley comes into contact with the colder air. The water vapor condenses around the vent opening, and drips back down into the cabinet.
When I have room left in the cabinet I will place a empty rack on the top tray, and something absorbent, such as cheese cloth, coffee filter, or folded paper towel on the rack directly beneath the vent. Sometimes I will use a small disposable 5" pie pan.