I had the problem this summer; it lasted for about a day, then went away.
This morning, suddenly, the Podcast app crashes upon launch. I have not updated to iOS 8; I still run 7.1.2 on an iPhone 5S. I have done nothing in past 24 hours except to sync with iTunes on my computer, which I do every day. Suddenly I cannot use the app at all.
I agree with others who long for the day when we did not have to use this piece of crap app to listen to podcasts. Podcasts worked fine in the Music app for me for years (and I've been listening to podcast for almost 10 years), but now that Apple has "improved" the scheme, it is garbage.
In my humble opinion.
I will be eager to read what others' experiences are like.
PS: Putting the phone into Airplane Mode does not seem to solve the problem for me at all.