The basic way to verify a Coach serial number is to call Coach at 888-262-6224 and give the customer service representative the bag's serial number, as of 2014. An authentic Coach handbag has a serial number registered with the company, and the representative can determine authenticity over the phone.
Having a serial number does not mean the bag is real, as counterfeiters put serial numbers on their bags as well. In general, the number should start with a letter from A to M.
If the bag does not have a serial number, it might still be real, but it could just be an older model. The owner can mail the bag to Coach to have it verified.