Will crab trap rust faster in the water or out the water?

Rust fastest when they are submerged.

After use, rinse & dry them before storing. If you keep them dry, they’ll last a lot longer.

If you are storing them near your dock out of the water I would think they would rust faster out of the water because of the salt air and increased oxygen. Was on a trip to Maine during the summer and noticed these metal pucks attached to the lobster traps and one of the fishermen told me they are pieces of zinc. The zinc acts as an anode and puts a small positive charge on the metal substantially slowing down the rust. Might be worth looking into.

I find out of the water is worse. Also if you have rebar in them put them opposite the hog rings makes a huge difference. Pressure wash them as little as possible. If you only have a couple traps I'd look into anodes.