I remember as a kid around this time being taught to "duck and cover" under our desks at school because the US feared a Russian nuclear attack. I think this was partly due to being on the west coast where an attack was thought to be most likely, but there was a lot of fear. Communism was spreading and was taking over other countries, thus Russia becoming the Soviet Union, Korea, Vietnam, etc. I think each side feared the other and wanted to blunt their influence in the world. It's also quite possible, knowing what we know now having seen the KGB files, that the USSR hoped to eventually take over the USA. They had a lot of agents here, and quite a few highly placed ones, although most were turned Americans not trained Soviets.