Take the AMTRAK from BWI to Washington DC, Union Station. The NE Regional Service leaves BWI towards DC every half hour, IIRC.
Amtrak - Stations - BWI Airport, MD - BWI Thurgood Marshall Airport (BWI)
When you arrive at Union Station, go to where the Metro rail platform is and take the Red Line towards Shady Grove. Get off at Metro Center and go to the Orange/Blue Line platform heading in the direction of New Carrollton (Orange) or Largo Town Center (Blue). Take either Orange or Blue line train and get off at the Smithsonian exit. That's it. When you travel on the Metro, you need to use a fare card. You can purchase one with various $ amounts at any of the Metro rail stations.