The current shutdown of the non-essential US government operations should not play too large a part in the weakness of the US dollar, and arguably is more of an embarrassment than anything else. However, the same can’t be said for the looming US debt ceiling, which could harm the currency badly. At the current rate, the US is expected to hit its limit of $16.7 trillion around the second week of October. If the two political parties allow this to happen, one likely consequence is that the country will default on some of its sovereign debt payments. This would almost certainly prompt some or all of the debt-rating agencies to consider downgrading US debt.
Having already suffered the loss of its AAA rating, the US would not want to see its status fall any further. The lower the rating, the higher the returns that need to be offered to lenders to compensate for the perceived increase in risk, and therefore the more expensive it would become for the US to borrow money. This would be a very slippery slope, and could easily cause the $16.7 trillion figure to spiral out of control.
We have seen the euro and other currencies begin to factor in this perceived risk, and we could see further dollar weakness the closer we get to the ceiling.