Straight from the source: Amtrak's bicycle policy

Unfortunately, none of those bicycle-rack-equipped trains run between NYC and DC. There is one train that stops at NYC and DC that has checked baggage service, which may allow you to check your bike as baggage: http://www.amtrak.com/servlet/Conten.../1237405732511

The other NYC-DC trains such as Acela Express or Northeast Regional don't have baggage service I believe.

Another alternative might be by bus. Bolt Bus do allow a bicycle as luggage according to their FAQ: https://www.boltbus.com/faq.aspx And they run between NYC and DC.

Hope this helps.