You either keep records yourself or you have a software package that does that for you.
If you have a script that does your backups, issue "mt -f /dev/nst0 status" (or whatever tape device you're using) just
before doing the backup and
again when it's done. Then grep for the string "Soft error count since last status=
N" (well, only search for the string up to the "=") in the result of the
second "mt status" command. If you see that
N is non-zero, that's probably a good-enough indicator that the backup is questionable and it's time to replace the tape.
It's good that you are thinking of monitoring that. My backup script was warning me about errors that occurred during the session. Next thing I know I'm ejecting the tape (manually which was odd since the script does a "mt rewoffl" once the backup is done) and part of it is hanging outside the cartridge and flapping in the breeze. Used that cartridge one time too many.