If your Explorer is equipped with the "factory tow package" (did it come with the 2" hitch receiver, and 7-pin plug installed?) your trailer battery is being charged whenever you are connected to the tow vehicle and the engine is running. Ford includes a relay in the tow package that sends voltage to the battery only when the key is on and the engine is running.
The trailer battery cannot discharge the vehicle battery when the engine is shut off. I've had 3 Fords with tow packages (92 thru 00) and each one worked that way.
Verify this information with your dealer.
If you don't have the "factory tow package" install an "isolator" between the vehicle battery and the trailer battery. That is a diode that only allows voltage to travel from the vehicle to the trailer. Cost is about $50 and you can install it yourself in about 1 hour. ;D