I finished all of mine with Varathane Spar Urethane, which is an outdoor water based urethane. It gives it a clear, tough, virtually waterproof finish. The caveat is that it takes a number of coats to do the job and all of the between coat procedures (light sanding etc.) that are necessary for a good job with any finish. The neat part is that this type of finish dries to the touch in a matter of minutes, so it doesn't pick up the dust particles like slow drying solvent finishes. And as I said, it is a clear finish that does not "amber" like solvent finishes. It is also very easy to repair.
I indicated that this finish is virtually waterproof. That is the case with most wood finishes--very few that I know of that are completely waterproof. Because of that, I also wax the paddles from time to time.
I am a wood worker and have used a lot of different finishes over the years. This water based urethane has turned out to be the best of all I've ever tried. That's not to say there isn't something new on the market that I have yet to discover, but I have had extremely good results with all of the furniture I build with this product. I have tried other brands, but have had the very best results with Varathane outdoor. It costs a little more than the indoor version, but I think it is worth it.
One more thing--be sure to use a very fine bristle, high quality brush.