If someone takes checks from you and tries to cash them but none are cashed if you the victim are not interested in pressing charges can the bank go after the person who tried to cash them?
Attempting to cash stolen and forged checks is a crime in and of itself. It does not matter that the crime was no completed because of the bank's diligence or the stupidity of the criminal. The District Attorney (and City Attorney where delegated) has the authority to make the decision of whether or not to prosecute, not the victim.
The bank did not lose anything and therefore does not have a civil claim against the thieves.