Enough coats of polyurethane should provide enough protection.

I'd probably use Waterlox.

Here is a link that might be useful: finish and moisture

Actually, the problem may be that only the front was finished and the back and bottom of the wood was not finished. Finishing just the front---especially without repairing the obvious damage will do little good.

"Actually, the problem may be that only the front was finished and the back and bottom of the wood was not finished. "

A distinct possibility.

While no finish stops water vapor, they only slow its movement it is possible to stop liquid water pretty well.

Small section pieces (especially) need to be finished on all side if they might be exposed to water more than accidentally.

If water enters from the unfinished sides it can easily try to escape past the finish, and in many cases till simply force the finish off the surface and produces cracks in the finish film.

The cracks then allow water to enter from the finished side and continue forcing the finish off the wood.

I routinely finish all sides of even quarter round or should molding used in kitchens.