This feature is disabled everytime the phone is turned off and turned back on for whatever reason I believe. Much like how you have to put in your passcode again to unlock your phone after your phone reboots. It is asking for a password to enable Touch ID again, if you read the popup box where you enter the password.
But yeah, make sure you do Settings>Touch ID and Passcode>iTunes and App Store>On
I believe you could workaround the disable after reboot by just going into the location in settings I described before, and toggling iTunes and App Store off and then back on. It will ask for your Apple ID password to enable Touch ID, and then you next download can be authorized with Touch ID.
I suffered from this same problem because I get *new* apps few and far between, and usually in that time my phone has restarted a time or two, so I would always need a password instead of Touch ID for apps.