Well, it's hard to separate the problems in the US economy from the problems in the world economy. But if I had to pick one of those, I would say the dollar is falling more because of the problems of the US economy than of the world economy.
Mainly, I think that the dollar is falling because fewer people want to buy US securities because they no longer seem to be such a good investment. This reduces demand for the dollar.
I also think that it has something to do with the loose monetary policy the US is pursuing at this time. The interest rate is very low, which also makes US securities less attractive.
At the same time, you should note that the dollar is actually up against the yen and flat against Chinese currency.