National Airport is in DC (actually in Virginia, nominally in DC), Dulles is in Virginia, and BWI (Baltimore Washington International) is in Maryland (closer to Baltimore than to DC).

National is the most convenient if your destination is DC and if you're coming from within the US (it's not an International Airport). Since its renovation in 1997 it's a delightful, friendly airport. We're right proud of it!

Library of Congress via Washington Airports

From National

• Metro. Yellow Line. Transfer to blue or orange at L'enfant Plaza, to Capitol South. Alternatively, you could take the Blue Line at National (both Yellow and Blue run on the same track) all the way to Capitol South and thus avoid transferring, but this will take much longer, as the Blue Line takes a circuitous route.
• Taxi. Should take no more than 20 minutes, even in rush hour. Tell the driver "Library of Congress Madison Building, Independence Avenue between First and Second, Southeast". If he doesn't understand, get a different cab. According to http://www.metwashairports.com/national/taxirates.htm, fare should be about $8.
• More information at http://www.metwashairports.com/national/ground.htm

From Dulles

• See http://www.metwashairports.com/dulles/ground.htm

From BWI

• See http://www.bwiairport.com/
  
  You can take a shuttle to the train station and take an Amtrak (800-872-7245) or a MARC Train (410-539-5000) to Union Station. It's about a 40 minute ride. MARC is for use only by those travelling to LC for MARC related meetings. (Just kidding. MARC is the Maryland Rail Commuter Service.) There is also a shuttle service .