Would like to avoid driving to and from Baltimore to Leesburgh, VA on weekends.

Walking would be cheapest.

I don't think there is any easier way than driving. I am assuming you mean Leesburg.

Weekends are better than weekdays.

Is this a regular trip or something?

IT will be convoluted--is your goal to avoid all driving? Also, do you actually want the cheapest because you can probably avoid a lot of driving but you will forfeit a LOT of time.

You can take the Marc Train from BWI to Union Station. From Union Station take the Metro Rail to Vienna or West Falls Church station and then either drive from there (not sure where you'll get a car) or take a commuter bus as scheduled here:

http://www.loudoun.gov/index.aspx?NID=228

Driving, annoying as traffic is, is going to be easier.

Why do you want to avoid it? The traffic in the area during the week can be a bear.

Dulles is a whole lot closer. Are the flights to Dulles that much more (including rental car cost)?
Under the best traffic conditions it will be 1.6 to 1.8 hours from BWI to Leesburg.
Dulles has a toll road (267) that goes into Leesburg.

Thanks to all for your replies.
Problem with flying to Dulles (apart from the extra expense), SW goes thru Chicago to get to Dulles. I'm thinking Marc to Union Station and metro to Vienna is my best bet.

You had better check weekend track work at www.wmata.com. You could be on for one hell of a trip.