It was largely our own governments wanting to scare us of nukes so we'd be scared of the Soviets, like in America with the schoolchildren doing duck and cover drills.

Having enemies the population is afraid of is good for politicians and they'll take any enemies they can find, and they'll do so indiscriminately regardless of the real nuance of the issues.

Immigrants, abortion, this religion or that, rock music, jazz music, alcohol, marijuana, global warming, windmills, books... just whatever as hard as they can regardless of if it's reasonable or not.

The Soviets were parking nukes in Cuba in striking distance of the White House. If that's not legitimately terrifying to you, I just don't know.

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.