MARC train now runs on weekends to Union Station as well. Cost is $7 each way I believe.
Either that, or the 35A(?) express bus will take you from BWI to the Greenbelt stop on the DC Metro rail that can take you downtown. Cost is $7 cash plus Metro fare once you get on the train.
Don't drive unless it's the middle of the day or if you need a car in DC. Do not underestimate DC traffic during rush hour, which could be 6AM-10AM and 3PM-7PM, and worse on weekends.