Welcome to Apple Support Communities. I understand that you're unable to enroll a fingerprint or enable Touch ID on your new iPhone. I know it's important for your iPhone to work properly as well as to be able to depend on the Touch ID for your security and I'm happy to help. I'd recommend to following the steps listed below. Be sure to try using Touch ID after each step:
Use Touch ID on iPhone and iPadMake sure that you have the latest version of iOS.
Make sure that your fingers and the Home button are clean and dry.1 For the Home button, use a clean, lint-free cloth to wipe off any dirt or debris.
Your finger should cover the Home button completely, touching the surrounding metal ring. While Touch ID is scanning, you shouldn't tap too quickly or move your finger around.
If you're using a case or screen protector, make sure it doesn't cover the Home button or the surrounding ring.
Go to Settings > Touch ID & Passcode and make sure that iPhone Unlock or iTunes & App Store is on, and that you've enrolled one or more fingerprints.
Try enrolling a different finger.
If you can't enroll any of your fingers, take your device to an Apple Retail Store or Apple Authorized Service Provider, or contact Apple Support.
Best Regards.