South Korea's Kospi index dived 4.1 percent but later recouped some losses to trade 3.1 percent lower at 1,782.50 by early afternoon. The Korean won fell 1.6 percent against the U.S. dollar, a traditional haven in times of uncertainty. The Japanese yen and euro also weakened against the dollar.
Japan's Nikkei 225 index was down 1 percent at 8,314.44, Hong Kong's Hang Seng slid 2.5 percent to 17,833.42 and the Shanghai Composite Index fell 2.6 percent to 2,167.68.
Kim Jong Il's death was announced Monday by state television from the North Korean capital, Pyongyang. It raises the possibility of increased instability on the divided Korean peninsula as the reclusive regime undergoes a leadership succession.
Those worries are most acute in South Korea and Japan, which have often born the brunt of North Korea's mercurial military and diplomatic actions.
South Korea put its military and police on alert and its president, Lee Myung-bak, convened a national security council meeting. In Tokyo, Japanese leaders said they were watching markets closely and in contact with the U.S., Kyodo News Agency reported.
"We need to prepare for any contingencies," Kyodo quoted Jun Azumi, the Japanese finance minister, as saying.
Kim, who had been ailing after suffering what is thought to have been a stroke in 2008, died at age 69 on Saturday.
Kim had presented his third son, the twenty-something Kim Jong Un, as his hereditary successor, putting him in high-ranking posts. But even with an heir apparent, some North Korean observers fear a behind-the-scenes power struggle or nuclear instability.
Markets in Taiwan, Singapore, Australia, New Zealand and Indonesia also fell.
"Particularly with the bearish market sentiment now, any negative news will make the market much more gloomy," said Kwong Man Bun, chief operating officer at KGI Securities in Hong Kong. The Hong Kong benchmark dipped 100 points after the news hit which "reflects concern over potential political instability," he said.
Kim's death overshadowed what already was a gloomy start to the week as jitters about Europe's debt crisis weighed on sentiment. Fitch Ratings said late last week it could downgrade the credit ratings of six European countries - heavyweights Italy and Spain, as well as Belgium, Cyprus, Ireland and Slovenia.
Meanwhile, Moody's Investors Services downgraded Belgium's credit rating and Ireland released data showing its economy is in worse shape than expected, with third-quarter GDP shrinking 1.9 percent.
Coming just a week after EU leaders struck a deal they thought would contain the continent's debt crisis, the onslaught of negative news shredded hopes of a lasting solution to the turmoil that is endangering the euro - the currency used by 17 European nations - and threatening the entire global economy.
"Everyone is waiting to see what comes from the next conference of European nations. Hopefully something good," said Jackson Wong of Tanrich Securities, in Hong Kong.
Chinese markets fell Monday after rebounding at the end of last week on speculation that the government might further ease reserve requirements for banks to help increase the amount of money available for lending to support growth.
"Everything came up empty" over the weekend, Wong said. "We are giving back the gains we had Friday."
Benchmark oil for January delivery was down 86 cents at $92.72 a barrel in electronic trading on the New York Mercantile Exchange.