In order to set up Touch ID on an iPhone, you first have to supply a password. So, possibly you are going about it in a way that is preventing you from setting up Touch ID.
Why not try it this way:
1. Select which options you want to use Touch ID for at the top of the screen
2. Type in the Lock Screen passcode that you wish to use in conjunction with Touch ID
3. Then try Adding your fingerprint at that point. Be sure to lift and touch the Home key as instructed. Lifting and placing your finger on the key is critical in getting the fingerprint to register.
4. Tap on Continue when you are asked to do so, and then continue to lift and touch the Home key by following the instructions to complete the fingerprint.
5. You can also try with a totally different finger. Maybe the one you are trying to set up the recognition for has an issue that is preventing it from completing.
Cheers,
GB