Long gun or pistol??
ONLY a FFL Dealer can MAIL a pistol through the US Post Office, but you can mail a long gun direct to a FFL licensed gunsmith.
This is the cheapest way and as long as you ship it Registered Mail and fully insured it's very safe.
If you're shipping a pistol yourself you can only use UPS and Fed-Ex.
Both companies have rules that you must ship the fastest (and most expensive) way.
This is so a gun is in their system as short a time as possible, so their workers won't steal it.
I think they'll ship long guns a little cheaper.
As long as you obey the Federal laws on shipping, you "should" be safe, given that you live in New York.
If you're worried about it, have it shipped by a FFL Dealer.
They can ship cheaper, but may scalp you on their fee for doing so.