There are several possibilities. One is to trust in the "bankruns" to check the issuing institution. The notes wouldn't be of that big value, maybe 1000 sat.

Another option is to mark each note with a bitcoin address, letting the receiver of a note check on the blockchain that the bitcoin is still there.  But that could only be used for big value notes. 

It will never be completely risk free, though. You should never have more bitcoin in notes than you can afford to lose.