i stole this from Rogers Answer Man:
Let's look at the problem: No matter if you walk or run, rain will fall on your head. This is unavoidable. If you run, you will spend less time in the rain, thus less total rain will fall on your head, and you will remain drier. However, if you run you will also be running into raindrops — that is, you will be hitting them with your chest as you proceed to your destination. If you walk, you cover less horizontal ground each second, so you run into fewer raindrops per second. (Looking at the extreme case, if you stand perfectly still, the only rain that will hit you will be the rain that falls on your head. However, you won't get out of the rain until the rain stops, so this usually isn't an option). Is it worth exposing your chest in exchange for getting fewer raindrops on your head?
Let's establish an "ideal" rain shower. There's no wind so the rain falls straight down, and the rain falls at a constant rate and density on every part of your path.
Clearly, the amount of rain falling on your head from above is directly proportional to the amount of time you spend in the rain. For instance, if you're out in the rain for 10 seconds and get hit by 100 drops, then if you're out for 20 seconds you'll get hit by 200 drops.
But what about the horizontal component? It turns out that this is directly proportional to the distance travelled. While it might seem that running exposes you to more rain on the front of your body, it really doesn't. You are exposed to the same amount — it just happens more quickly.
With our idealized, constant rain, the density of raindrops in front of you is always the same; some of them are falling out of the frame, but new ones are coming down to replace them. So whether you walk through the rain or run through it, you will pass through the same volume of water. Imagine if you could freeze the scene with the rain suspended in the air — the water you see in front of you is the water you have to walk "through," and whether you move through it slowly or quickly, that's the amount of water you're going to hit.
So the vertical component of the rain is proportional to the time spent in it, while the horizontal component is proportional to the distance travelled — and the distance is the same in both cases. Obviously, if your sole motivation is to stay dry, it always makes more sense to run.
If you factor in wind, there are some slight changes to be made (if the wind is at your back, it's best to move at the same speed as the wind — not slower or faster), but running is inevitably the drier method. There's an amazing calculator here that shows you exactly how much drier you will be if you run. It lets you factor in wind speed, the speed and angle of the rain, the size of your body, and the distance to your location. Plug in some numbers and see how it comes out.
There are a few other issues you might want to consider — if you're travelling through standing water or puddles, running might cause splashes that walking won't, and if you're running you might be more likely to fall and get really soaked. But overall, running is the best method.
Sources: Seattle Times, Ask Dr. Math, Great Moments in Science