What to do if I paid an individual to install gutters and he has cashed the check but after 5 months has failed to do the work?

It is not a crime in New York unless the State can prove that he never intended to do the work, which will be very difficult to do.  The police may view this strictly as a civil case and advise you to sue them.   On the civil front, you can file complaints with the Better Business Bureau and if the individual is licensed, with whomever the licensing agency is.  If those complaints fail, you can still sue them for the money that you lost.  If it is under $5000, your best bet and cheapest way to do this is to bring an action in Small Claims Court.

You may be able to have him arrested and probably should report this to the police--if he took and cashed the check with criminal intent (e.g. with the intention of never doing that the work), he may be subject to prosecution. However, that will not necessarily get your money back--prosecution and the criminal law are designed to punish wrongdoers, not reimburse their victims, and reimbursement is often not a part of criminal prosecution. Therefore, you should also consider suing this person to recover  your money; you have the option of suing in small claims court, where filing fees are low and you can act as your own attorney, to reduce the costs of a lawsuit.