The water was mainly used to cool damaged reactors at the Fukushima power plant north of Tokyo, after it was hit by an earthquake and tsunami in 2011.
The release of the water from huge storage tanks into the Pacific is expected soon though no date has been set.
Japan has given repeated assurances that the water is safe, saying it has been filtered to remove most isotopes though it does contain traces of tritium, an isotope of hydrogen hard to separate from water.
But fishermen and shoppers in Japan and across the region are afraid.
"I recently bought 5 kilograms of salt," Lee Young-min, a 38-year-old mother of two children, said as she made seaweed soup in her kitchen in Seongnam, just south of the South Korean capital, Seoul.
She said she had never bought so much salt before but felt she had to do what she could to protect her family.
"As a mother raising two children, I can't just sit back and do nothing. I want to feed them safely."
The rush to stock up contributed to a nearly 27% rise in the price of salt in South Korea in June from two months ago, though officials say the weather and lower production were also to blame.
In response, the government is releasing about 50 metric tons of salt a day from stocks, at a 20% discount from market prices, until July 11, Vice Fisheries Minister Song Sang-keun said on Wednesday.
South Korean fisheries authorities say they will keep a close eye on salt farms for any rise in radioactivity. South Korea has banned seafood from the waters near Fukushima, on Japan's east coast.
China has also criticised Japan's plan to release the water, accusing it of a lack of transparency and saying it poses a threat to the marine environment and the health of people around the world.
Japan says it has provided detailed and science-backed explanations of its plan to neighbours.
Japanese Chief Cabinet Secretary Hirokazu Matsuno said last week Japan was seeing increasing understanding on the issue though that was not so apparent in Seoul shops this week.
"I came to buy salt but there's none left," said 73-year-old Kim Myung-ok standing by empty supermarket shelves. "There was none the last time I came too."
"The release of water is worrying. We're old and have lived enough but I worry about the children."
