The traffic in Washington DC and surrounding area is horrendous, especially during rush hour - and unfortunately, the traffic between Dulles Airport and the the Beltway or downtown Washington is among the worst in the DC area. There's really just one main road between the airport and the Beltway. And of course Dulles is just over 25 miles (40km) from downtown Washington, so even in the best of circumstances it will take a while. Plus they've been doing construction in that area for years, and won't finish the construction for a couple more years. It seems like any time I drive to or from Dulles Airport (which isn't too often), the traffic patterns change because of the construction and the signs are not well placed, so it gets confusing. Public transit will be less stressful than driving for sure. I know it's expensive, but parking downtown can be as well. And it will probably take the same amount of time (especially considering it can take a while to actually find parking downtown). The easiest way to do this drive is to take the Dulles Access Rd. (I believe it's State Route 267, and keep in mind this is a toll road) east to Interstate 66. Follow this directly into Washington DC, crossing over the Theodore Roosevelt Bridge. As soon as you cross the river into Washington, you can either take Constitution Ave. and find parking right near the National Mall, or continue up the Whitehurst Freeway to K Street and find parking near there. If you park at a garage, you'll probably spend $10-15 a day for a weekend and twice that on a weekday to park; finding legal street parking isn't easy but it does exist.
If you can change where you are staying, I would look at hotels in Crystal City, Virginia. They're just a couple of miles from downtown, a much quicker and cheaper Metro trip, and there are probably a dozen hotels within a couple blocks of the Metro. They're pretty affordable and I can't imagine they're much more than hotels near Dulles, but even if the hotel costs a few dollars more a night, you'll probably save at least as much in transportation costs and a LOT more time and stress.