Oil based paint or stain would be better. Even something like painting with linseed oil would work fine. You don't have to "paint" the wood per say, but the goal should be sealing it. Sealing will make it last longer and easier to clean. An oil based paint wont flake and come off as easy as a latex paint. I think oil based paints adhere to wood surfaces dramatically better. But either oil or latex based would be fine.