As you walk out of the way of one raindrop, you're moving into the path of another one. This means the amount of rain that's hitting the top of you is constant, regardless of how fast you're going.
You can picture that the raindrops themselves are stationary, and you - and the Earth below you - are moving upwards through the rain. And since the volume of a parallelepiped - a 3D parallelogram - doesn't depend at all on its slant, then no matter how fast you're moving horizontally, the same amount of rain will land on top of you each second. Which sounds like a pretty obscure comparison, but when you see it illustrated, it makes a lot of sense.
This means that if you're not moving, the rain that's falling on top of you is all you'll get. But when you start moving, you'll also come into contact with raindrops from the side. So really, if you want to stay driest, you should just stand still, in the middle of the rain. Of course, this isn't exactly practical, and people will probably drive past and think you've gone mad. So you have to move, and just like how a snowplough will drive away the fallen snow in its path no matter how fast it's going, the amount of rain you'll come into contact with, whether you walk or run, will actually be the same. Because, again, parallelepipeds.
Over a given period of time, the same amount of rain will hit you from the top, regardless of how fast you're going. And over a given distance, you'll hit the same amount of rain from the side, again, regardless of how fast you're going.