The 15 organisations which include WWF Switzerland, Greenpeace, Fokus Anti-Atom and various chapters of the Green Party, noted that Oldbury in England, inaugurated in 1967, will shut down next week.
They said Beznau should also be shut down.
"Many security problems show that Beznau has run its course," said the organizations in a joint statement about the nuclear plant that began operating in 1969.
They said there are cracks in the mantle of the reactor and in the steel containment shell, something strongly denied by Axpo, the energy company that operates Beznau.
"As a precaution the lid of the reactor is to be changed, but there is no crack," said an Axpo spokeswoman.
Beznau is scheduled for decommissioning in 2019 after 50 years of operation.
Last September, the Swiss Parliament approved a nuclear phase-out for the country's five nuclear reactors, due to be decommissioned by 2034.