First thing to check is the plug itself. You didn't give any kind of details about what kind of trailer wiring you have, so its hard to make a guess. If, for example, you have a flat 4-pin plug, the contacts are made out of tinned copper. Those contacts tend to deteriorate quickly when exposed to salt unless you keep them well greased. That would be the first thing I look at.
If you have a round socket connector, check the wires leading into the back of it. Often water can get in there and damage the wires.
Now not sure if you have factory or aftermarket wiring.
If you have factory wiring, there should be a plug about 2-3 feet back from it at the driver's side frame rail. You can check there for continuity on the line that is giving you problems. If you find power there, you may need to replace just the tail portion of the wire. Also check the trailer fuses and relays under the hood.
If you have aftermarket wiring, make sure that your normal lights on the truck work properly. If they do, replace the aftermarket unit since its probably shot.
Another simple test for you before going crazy: Do the BRAKE lights work on both sides? If they do, then the problem is in the trailer light converter or the brake signal running to the converter.