This also started to happen to me today, just today. iPhone 5S, iOS 7.1.1, Podcasts version 2.1.2.
I had one podcast (Le Show, by Harry Shearer) that was giving me trouble; it hadn't successfully downloaded either to my laptop nor to my iPhone in the past three weeks. Last night I deleted it, thinking I was probably hooked up to an older, problematic feed. I resubscribed in iTunes, and everything seemed to work fine. I was able to get two recent episodes, and (I thought) sync them to my iPhone
Just now I got in my car to run some errands. That podcast wasn't there. I came back into the house to sync again, and when I launched Podcasts on my phone, it crashed as soon as I tried to make any selection from within the app. That is, it opened up okay, but as soon as I touched the screen to make any menu choice, it crashed.
Turning off WiFi seems to have cured that problem. Of course, that makes the iPhone useless for other things, doesn't it.
Why did it start automatically happening to us all today? I presume, like me, none of you have updated either iOS or the Podcasts app today, right?
Creepy! and Frustrating!