MP3 players can certainly be used for jogging and other high-impact activities. However, it would be a good idea to get a flash-based player, rather than a hard-drive one. These don't skip because they have no moving parts, and they are also available in very small sizes.
As to what I would recomment, Apple has just released the second-generation iPod Nano, which you can get for $219 (2GB) or $299 (4GB). If you want to get serious and tech up your running, there's an Apple/Nike accessory for this player that works as a pedometer and tracks your progress with software. It's being released in October, but you can see our review here.
Another sport-focused player that is about to be released is Sony's NW-S200, for $249.95. This model also has an FM radio and comes with an armband; check it out here.