Expectations that the shutdown will hurt the economy and prompt the Federal Reserve to postpone the start of its withdrawal of monetary stimulus weighed on the US currency.
Previous shutdowns have ranged from a day to nearly a month.
After trading weak in Asian hours, the dollar's selloff gathered pace in European trade with the index falling to 79.864, its lowest since Februrary 13 and down 0.4 per cent on the day.
It was last trading at 80.056, still down 0.2 per cent.
The dollar's weakness lifted the euro to an eight-month high of $1.3589.
Hedge funds bought the single currency, which was also helped by the prospect of Italian Prime Minister Enrico Letta's coalition government surviving a confidence vote on Wednesday.
It was last trading at $1.3550, up 0.2 per cent, shrugging off a rise in German unemployment.
Eurozone unemployment also remained stubbornly high at 12 per cent.
The Swiss franc hit a 1-1/2 year high of 0.89925 francs per dollar on trading platform EBS.
Speculation the U.S. shutdown could prompt an early release of the closely watched monthly US jobs report added to uncertainty in financial markets.
Officials said last week that the U.S. data will be delayed.
Reflecting the nervousness, near-term implied volatilities, a gauge of how choppy a currency is likely to be, rose.
The one-month euro/dollar implied volatility rose to 7.5 per cent, from around 6.6
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