This is a bad idea, as one can change a compromised password as many times and necessary or desired.
Assuming a print from a single digit is enough, you're limited to ten total passwords without starting to leave the realm of social acceptability. On top of that, this uses only a public, nonsecret method. It's not combining something that you have with something that you know, preferably something known only to you, and since it's from a read-only source, once it is compromised you're screwed.
If some biometric system is used in concert with a strong user-selected bit of information, like a password, passphrase, or numeric string, then maybe it'll be okay, especially if the system does not indicate to the user where the failure in authentication happens (ie, confirm that one has the right fingerprint before rejecting the password). If the fingerprint is used as an analog for the user id, and the password is still one's personal secret, that may work.
If the issue is PINs being commonly four digits long, people have demonstrated an ability to remember ten-digit numbers as many markets now have ten-digit dialing for local calls with several area codes. I don't think that it would be an undue burden to use PINs longer than four digits in this age on account of that. What would be best is for there to be a minimum length that's greater than four or five, but a max possible length that would be well larger than most users would need, so those who do want longer credentials can use them, and with all of the number of places in between also being supported.