Is it illegal to send prescription medication internationally to New York State in the US?

It depends on the medication. It is definitely illegal to have prescription painkillers mailed (because they are scheduled drugs). Codeine, in all forms, is scheduled and controlled in the US. Normal medications, like antibiotics, are in a different category and are not really a problem. There is even a law in the US that customs isn't supposed to seize non-scheduled medications, as along as they are in individual patient quantities.