A short-circuit or bad connection in the connector should not cause the LED lamp to break. It probably broke for some other reason. But you'll probably want to deal with the corrosion anyway, as it is easier to do so now before it gets too bad.
First off you'll want to clean off the corrosion that has already happened. A mild acid is a good option, you can read more in this question.
As for sealing, my recommendations would be:
1) Plain old grease (vaseline, silicone spray or even chain lube if you have nothing else). The metal parts will push it aside and make a connection, but it will stay elsewhere to keep water from getting into the area. There are also special greases for electrical connections, which probably work even better.
2) Self-vulcanizing rubber tape. Wrap it around the connector, and it will vulcanize into a seamless tube that stays in place better than normal tape.