The two airports are in competition, and as far as I know, public transportation between BWI & IAD is not direct. One can get to DC by train. Then to Vienna Va by metro, and to the airport by Flyer. May take all day to do this. Taxi can run $100 with a little traffic.

Dulles has Greyhound service, but I don't know if it goes to DC.

Rent another car for an hour, and drive.