"The European Central Bank failed to relieve investor concerns about the progress of the debt crisis," said Christopher Vecchio, currency analyst at DailyFX in New York.
Draghi's comments came as the ECB predicted the euro zone economy would shrink again in 2013 and sharply lowered its growth and inflation forecasts. It also said risks to growth remained on the downside.
The bank forecast gross domestic product in a range of falling by 0.9 percent to growing just 0.3 percent next year, suggesting contraction was far more likely than not. It predicted inflation at 1.1 percent to 2.1 percent next year.
"The combination of the ECB's cooler growth and inflation forecasts opened the door to a rate cut in the months ahead," said Joe Manimbo, senior market analyst at Western Union Business Solutions in Washington.
The euro fell as low as $1.2948 on Reuters data and was last down 0.8 percent at $1.2966. At current levels, it is on track for the biggest daily percentage fall since Nov. 2.
Some US$6.04 billion in euro changed hands, according to Reuters Dealing, the highest daily volume since Oct. 9. The euro lost 0.9 percent to 106.80 yen.
Italian and Spanish government bond yields rose on tensions in Italy. A disappointing Spanish bond sale on Wednesday also weighed as it revived talk of an official bailout request from the euro zone's fourth-largest economy.
"Some analysts are predicting Italy could be contracting by as much as negative 3 percent next year. If that's the case, then they could become another financing crisis for the euro zone," said Boris Schlossberg, managing director of FX Strategy at BK Asset Management in New York.
Investors are shifting attention to Friday's closely watched government report on U.S. nonfarm payrolls for November. Payrolls likely rose only 93,000 last month after advancing 171,000 in October, according to a Reuters survey of economists. The unemployment rate is seen holding steady at 7.9 percent.
"After selling off as a result of comments coming out of the ECB conference, euro/dollar has found support around the 1.2970s, which is very close to key support of 1.2950," said Marc Principato, Director of SMB Forex Trading And Education in New York.
"I am looking for price stability around this level as we await the NFP report tomorrow." Some analysts said that while the outlook for the euro zone remained bleak, euro losses against the dollar could be limited.
"As we expect the (Federal Reserve) to announce a shift to outright Treasury purchases next week, the U.S. dollar should remain soft," said Vassili Serebriakov, currency strategist at BNP Paribas in New York. The Fed holds its December policy meeting next week.
BNP forecasts the euro at $1.33 by the end of the year. The dollar slipped 0.1 percent to 82.37 yen, still not far from the 82.82 touched on Nov. 22, which was the highest since early April.
Traders expect the yen to remain under pressure on expectations of further monetary easing by the Bank of Japan following an election on Dec. 16.
The Australian dollar rose to a 2 1/2-month high after surprisingly strong Australian jobs data prompted investors to reduce expectations of further policy easing. It was last up 0.2 percent at $1.0476.