Any of the area airports would work (assuming you are not renting a car, if you were renting a car, White Plains would by far be the easiest way to go).

By public transportation, you could get to Grand Central, either taking the airport bus or the airtrain subway train combinations. Exactly how depends on which airport. From Grand Central you would buy a ticket and take the Metro North Commuter train to Stamford.

They run quite often during rush hour, once or twice an hour not at rush hour, last train at 1 am or so.

Another option is the shared buses from the airport to Stamford,

http://ridethedot.com or http://ctlimo.com is like $60 or so I believe (each way), though call to see what airports and what the service is.

One of the limo bus deals would be much easier, you load your stuff at the airport and they take you to your hotel in Stamford. It costs more, but it's worth it.