Q: Touch ID on iOS 8.1.2

Try restoring iOS using iTunes on your computer.

Did you try to restore the iOS? Did it work?

Then most likely its a hardware problem.

If your device has a hardware problem, (if you have insurance) Your carrier should be able to replace your device for free as long as there is no physical damage or water damage! Maybe Apple would fix it for you if you called and sent it in.

If you have properly restored it as a new device [Use iTunes to restore your iOS device to factory settings - Apple Support], and Touch ID is not working [Use Touch ID on iPhone and iPad - Apple Support], then it's likely a hardware issue.  Get it serviced/replaced by Apple.

AppleTechDenver wrote:

 

If your device has a hardware problem, (if you have insurance) Your carrier should be able to replace your device for free as long as there is no physical damage or water damage! Maybe Apple would fix it for you if you called and sent it in.

Whether they have insurance or not with the carrier, they still have hardware coverage through Apple.