Unlike Samsung Android phones, there is no straightforward way to enable USB debugging on the Verizon version of the HTC One. However, it is not too difficult and it does not messes with the warranty.
To enable USB debugging, go to Settings > About > Software Information > More > Build number – tap Build number 7 times to enable developer mode. You will now get a window on the screen saying "developer mode has been enabled".
Go back to Settings’ main menu and under System section, you will have Developer Options added. You can easily enable USB debugging and many other things such as Background Process Limit under the Advanced section.