My wife just experienced the dreaded Podcasts app crash on her iPhone 5 with iOS 7.1.2. No panic, you can simply follow iMore instructions to solve the problem:

1. Pull up from the bottom of the screen for Control Center.
2. Tap the Airplane mode icon to turn of the internet connection (presumably to prevent Podcasts from trying to contact iTunes).
3. Pull to refresh on the main Podcasts screen (presumably that clears whatever bad bits got stuck).
4. Pull up from the bottom of the screen for Control Center again.
5. Tap the Airplane mode icon to turn the internet connection back on.