I think we generally just say “to Washington” in most of the country. In the northeast cities, you might hear “down to” about as often. (Ah, I see ErinPuff confirms that.) But in the southeast, you won’t really hear “up to Washington” so much.
Where I live and work, just a few miles away from DC, we actually say, “I’m going into Washington. (Wish me luck)”.
Ohioan checking in. Washington, D.C. is to the SE of here. In my experience, most people will say “down to Washington” or simply “to Washington.” For those who say “down to,” it’s a simple description of direction, not an indication of revulsion with the national capital or its officialdom.
Many thanks for all the replies. I think I’ve got a handle on national and regional usage which is what I was looking for.
I hadn’t thought of the railway system. I’m not sure it’s relevant but I used to be a trainspotter, for which activity please forgive me. I was very young at the time.
I wouldn’t know. Nobody in their right mind ever goes there. 
Seriously, it’s purely based on the compass. I live in Sydney, and would go down to Canberra, Melbourne, or Hobart, up to Brisbane or Darwin, and over to Perth. I have heard of some parochial Australians pointedly using over to where it would more sensibly be up to, but that’s rare.
The only exception is the nations railways, where UP trains always go to the state capital, and DOWN trains go away from it. When I went to Melbourne by rail last week, my train was going down until it crossed the Victorian border, and then it started going up.
Wait, you’re traveling to your money? You have so much you have to store it off site?