Seven people were saved after the 1,753-tonne Oryong 501 operated by South Korea's Sajo Industries went down in the western Bering Sea at 0520 GMT, according to the Ministry of Oceans and Fisheries.
The trawler was carrying one Russian inspector, 11 South Koreans, 35 Indonesians and 13 Philippine crew members, a ministry spokeswoman said.
One Korean sailor was confirmed dead while the Russian and six foreign crew members were rescued, she said.
"Several vessels from nearby waters were involved in search and rescue efforts, but the operation has been hampered by bad weather."
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Four Russian fishing boats are taking part in the search in stormy weather, Russia's Interfax news agency reported.
"The search is continuing. But it is starting to get dark in the sector and there has been a storm. Waves are six metres high, which is complicating operations," Artur Rets, in charge of sea rescue in the Kamchatka region, told the agency.
Russian authorities gave different figures for the number on board the Oryong 501, putting it at 62.
South Korea's Yonhap news agency said the sunken trawler was about 36 years old.