The US dollar is the only currency you can pay US taxes with and the only currency the US government issues debt in, so as long as the US government exists, there is a guaranteed demand for US dollars.
I know, I know, "But the US is about to implode any day now!" And if it does, the entire world economy will go with it. BitCoin depends on the Internet and the Internet depends on a functioning economy. BitCoins won't do you any good if your ISP and your power company are bankrupt.
So in any realistic scenario where BitCoins have value, so does the US dollar. However, it's entirely plausible that BitCoin will fail but not the dollar. Congress could prevent law-abiding businesses from dealing in BitCoins, shutting down the major exchanges and effectively isolating BitCoin from the traditional financial system. The illicit market would still exist, so you'd still be able to convert cash to BitCoins to buy drugs with... but most people will just buy drugs with the cash directly.