U.S. stock index futures fell, with the S&P 500 indicated to open down 0.4 percent ESc1 1YMc1 after share prices fell in Europe and Asia.
The Swiss franc, by contrast, was on track for its biggest single-day rise against the euro in more than 2 1/2 years.
"Trump's comments about North Korea have created nervousness and the fear is if the President really means what he said: "fire and fury"," said Naeem Aslam, chief market analyst at Think Markets in London. "The typical text book trade is that investors rush for safe havens."
Trump's remarks on Tuesday that North Korea would face "fire and fury like the world has never seen" pushed Wall Street lower on Tuesday and drove up the VIX "fear gauge" of expected volatility on the S&P 500 higher.
The VIX rose further on Wednesday, rising as far as 12.11, its highest in almost a month.
A spokesman for the Korean People's Army said in a statement on Wednesday it was "carefully examining" plans for a missile attack on the U.S. Pacific territory of Guam, which has a large U.S. military base.
In Europe, the pan-continental STOXX 600 index fell 0.9 percent, with falls deepening after a car rammed a group of soldiers in Paris, injuring six, in what officials said was a suspected terrorist attack. France's CAC dropped 1.6 percent and Germany's DAX fell 1.3 percent.
Tokyo's Nikkei 225 share index closed down 1.3 percent at its lowest since June 1 as the strong yen hit exporters, while South Korea's KOSPI index fell 1.1 percent to seven-week lows. South Korea's won currency dropped 0.9 percent against the dollar to its lowest close since July 13.
MSCI's main index of Asia-Pacific shares, excluding Japan, was last down 0.6 percent. Chinese blue chips closed flat but Hong Kong's Hang Seng fell 0.4 percent.
Instead, investors turned to assets that tend to benefit in times of geopolitical and financial stress.
The Japanese yen strengthened by 0.5 percent to around 109.70 per dollar. Japan is the world's biggest creditor country and there is an assumption investors there will repatriate funds in a crisis.
The Swiss franc reversed a two-week losing streak and gained 1.1 percent to as firm as 0.9611 per dollar The Swiss currency was also on track for its biggest daily gain against the euro since the Swiss National Bank removed its cap on the currency in January 2015. It was last up 1.2 percent at 1.1305 per euro.
"Heightened geopolitical risks overnight have seen the markets flip from risk-on to risk-off and we have to wait and see how long this move runs before adding some positions," said Viraj Patel, an FX strategist at ING in London.
The dollar index, which measures the U.S. currency against a basket of major peers, slipped 0.1 percent as U.S. Treasury yields fell.
The euro dipped 0.1 percent to $1.1733 but the single European currency has been slipping this week against the dollar, having hit a more than 2 1/2-year high of $1.1892 on Aug. 2.
Yields on core government debt fell. Ten-year U.S. yields dropped 4.3 basis points to 2.24 percent and German equivalents fell 3 bps to 0.43 percent, a six-week low.
Gold rose 0.6 percent to $1,268 an ounce "The market hates uncertainty and that's certainly what we have now," said Ole Hansen, head of commodity strategy at Saxo Bank. "But looking ahead unless we start to see a conflict break out or a major stock market correction, (gold) is capped at 1,295 (although) the upside at moment is the favoured direction."
Oil prices rose before a report expected to show U.S. crude stocks fell for a sixth week. Bent crude, the global benchmark, rose 19 cents to $52.33 a barrel.