I always am amazed by people suggesting that walking in the rain keeps you dryer than running. Just saw an answer to this. Check it out, it's nice:
I have also seen it debunked experimentally, by MythBusters. But let's try a different approach: intuitive math. Intuitive math is tricky because it usually is wrong, but hey, it's fun.
Apparently, we all agree that how wet you get correlates to your speed. Otherwise, the question is pointless because the answer is "walk or run, but take an umbrella", while true, is cheating, right?
So, for those slower-is-better proponents: go and walk very, very, very slowly. You may notice that you end completely soaked before you finish walking. If you didn't, you are still walking too fast.
On the other hand, if you were to go at 1000000 km/h we all agree you would only get some drops in your frontside, right? Which would not soak you. Right? And most importantly, is constant regardless of your speed, because it's just the average amount of water contained in a man-shaped prism from point A to point B, and you get that water in your front if you go slow anyway.
Assuming the speed/soakiness curve is roughly monotonous, it's clear that the maximum soakiness is when you go slowest.
If it's not monotonous, then the question is roughly unanswerable, since it would involve there is an optimal speed and it's worse to go either faster or slower than that, which means the answer is something like "jog" which is not what you want.
So, go fast, go dry.