Contact the developer/go to their support site if only one app.
Restore from backup. See:
Restore to factory settings/new iPod
Here's the reply I received today from OneClickDigital Support:
We apologize for the difficulty you have experienced recently while using the OneClickdigital iPhone App. The recent Apple iOS 7.1 update has caused our App to error upon playing. Our Development Team is working diligently to correct this issue. We do not have a time frame when this will be corrected. We will notify you when this issue is resolved.
Can you please provide the name of the library you are using to access the service for our records?
We apologize for any inconvenience.
OneClickdigital Support Team
This appears to now be solved with the release of OneClickDigital 1.3.2. The release notes reflect the addition of support for iOS 7.1, and I have confirmed that audiobooks now play correctly.