I think it came from peaceniks and hippies mostly. You're talking about the equivalent of modern anti-vax liberals. Anti-science and given to conspiracies and mysticism.

There was a pretty good reason to be scared of nukes when these folks were children in the 50s. The world was quite a different place back then. The US was lagging behind the Soviets, militarily speaking, and Communism was much more expansionary.