Italy, South Korea and Iran posted sharp rises in infections over the weekend. South Korea now has more than 760 cases, Italy more than 150 and Iran 43 cases.
The World Health Organization is now worried about the growing number of cases that have no clear link to the epicentre of the outbreak in China.
"If you expect global growth to be lower on the back of this ... it still makes sense to bet on a stronger dollar," said Morten Lund, an analyst at Nordea, also citing the US economy's relative strength and its higher yields over other developed markets.
The euro weakened 0.3% to $1.0805, close to last week's level of $1.0778 - a nearly three-year low.
"I expect euro/dollar to move lower in the short term, not only because of the incidents in Italy but because the euro zone is much more exposed to China (than the United States)," Lund said.
The dollar index gained 0.1% to 99.562, remaining close to its highs of 99.915 touched last week.
The Australian dollar, often traded as a proxy for China risk because Australian exporters depend heavily on Chinese demand, weakened 0.5% to a new 11-year low of $0.6585. The New Zealand dollar dropped 0.6%.
Currencies closely linked to global growth and commodity prices were also sent lower, with the Swedish crown and Canadian dollar down sharply.
The Chinese yuan fell in offshore markets. The dollar was last up 0.1% at 7.0398 yuan per dollar.
Emerging market currencies were also whacked lower.
Yet risk-aversion, which saw stocks tumble and gold rise, offered little support to the yen.
After partially recovering last week's drop on Friday, it traded marginally higher at 111.42 per dollar as Asian investors discount its safety value owing to Japan's virus exposure.
"The market reaction to the coronavirus appears to be evolving, beginning to differentiate the currencies vulnerable to the virus from the rest," Barclays analysts said in a note. The Swiss franc was unchanged against the euro at 1.0606 francs, very close to its strongest since 2015.