If you want the long, very interesting explanation of how to stay dry, you should watch all 18+ minutes of this video. It's worth it not only for the mathematical explanation but also for the wonderfully charming MIT professor who's explaining it and his hilarious jokes. (Great math crack at the 15:32 mark!) His accent could not be more perfect, his explanation more thorough, or his enthusiasm more infectious. He had me at "I'd now like to discuss a classic problem of Mary and Peter arguin' in the rain."
If you want the short answer: The faster you run the less wet you'll get, and if you lean forward while you run you're even better off. If you run infinitely fast, you won't get wet at all!