The area's public transportation will work perfectly for you.

If you were to go directly between the airports, the quickest would probably be to take the BWI airport shuttle to the MARC Penn Line/Amtrak station, either MARC or Amtrak to Union Station, the Metro Red Line to Gallery Pl/Chinatown, and the Yellow Line to DCA.

If you happen to board an Amtrak train continuing south of DC, you could take that past Union Station to Alexandria and then backtrack from there to DCA on the Yellow Line, but that may actually be slower than using Metro from Union Station, depending on the Amtrak train's layover at Union Station.

IIRC, there's also a cheaper way to do it involving a bus from BWI to the Greenbelt Metro station, but it won't be as fast and is probably more confusing.

Of couse, if you're not leaving until the next day, you'll probably end up staying somewhere overnight, in which case you can see other options for getting out of BWI and into DCA at http://www.toandfromtheairport.com.

I assume a moderator will be moving this shortly to the DC forum, as that is probably the more appropriate place for this.