Many doors will leak, not at the door or gasket, but rather from the mounting flange. When you consider the age of your RV, it would be a good idea to check closely the sealant between the trailer wall and the mounting flange on the door assembly.
I'd almost bet that there's some sealant that has dried out or some damage to the seal between the two components. You could simply apply some caulking around the surface and hope for the best, but I'd suggest removing the door assembly, cleaning all the old caulking from both surfaces and reapplying a quality BUTYL tape caulk to the door flange and then reinstalling it with the original screws. Don't forget to clean them as well and replace any that are rusted or damaged.
While you're at it, now would be a great time to climb on the roof, check all the DICOR seals as well as the condition of your EPDM roof.
Also remember that if one door has started leaking, the same may be happening with other storage doors or the windows. They are mounted to the RV the same way as the doors. Check carefully, many window leaks don't show any water intrusion around the window, but rather several feet away where the water pools inside the wall causing damage to the floor/wall surface from the inside out.
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