The only train option is Amtrak, which runs from South Station or Back Bay Station to Penn Station in NYC. But the buses, which all go from South Station or Back Bay Station to various spots in NYC, are fast, clean (mostly), and dirt cheap if you purchase in advance. Try boltbus.com, megabus.com (which leaves from Back Bay Station), gotobus.com, busjunction.com.
not sure about train but there are A LOT of busses that go from south station (boston) to NYC. For example, Fung Wah Bus (about $20). http://www.fungwahbus.com/
south station