Interestingly, I'd say the most likely part of you to get wet on JP is your face!
Same as with other posters above, I have ridden it and gotten nothing and I have ridden it and gotten fairly wet (I wouldn't say drenched, but I don't doubt it can happen).
I will say that I feel like I have successfully ducked to the side a bit for the splashdown (as long as I wasn't on the ends) and minimized the actual water in my face - similar to blana's suggestion above. But it is an improvement, not a solution.
As an additional note, the Popeye ride is a TOTAL DRENCHER. There is NO chance of not getting wet on that ride, IMHO. Though might be a little less "one big plash to the face" than JP is.
One last note - There's a definite risk of losing a hat (or face mask) on the JP ride. The end creates a pretty big wind and splash. Make sure if you go that route it is something that ties on or he holds it. I have first hand seen people lose caps on that ride twice - and I can't have ridden it more than maybe 10 times.