Americans' distrust of nuclear reactors began in part because of a movie called The China Syndrome that came out the same year as the Three Mile Island meltdown . The movie depicts nuclear reactors as much less safe than they actually are, and even though little to no harmful radiation escaped containment at Three Mile Island, the hippies got freaked out. Nuclear energy became conflated with nuclear weapons because of of what you described, which didn't help things. Actually the reason for the meltdown was that most of the technicians were trained on Navy reactors which have different safety protocols.

Modern reactors are safer than older pressurized water reactors like Three Mile Island, and much much safer than graphite-moderated RMBK reactors like Chernobyl. I don't know a whole lot about boiling water reactors like the ones at Fukushima, but my understanding is that they're usually safe as long as you don't put them in a tsunami zone with the backup generators in the basement.