I was in the Air Force at the time and was stationed in Alaska where one of our prime missions was to support Chrome Dome.
We got President Kennedy's announcement before the general public was aware of the situation and I thought the world was about to come to an end. We weren't that far from Russia and I figured we'd be one of the first bases to see the beginning of the end.
Everyone was put on Red Alert. It was pretty scary and no one I knew thought the Soviets would ever back down. After a couple of weeks, things settled down a bit and we were allowed off the base at least to go to town. On a bus ride to town I had the opportunity to talk to one of the Army guys that manned one of the missle batteries that guarded the base. I asked him what they would do if a barrage of missles came our way. He refused to give me any details, but he said, "Don't worry about it, we've got ya covered." I don't know if that made me feel much more secure, but at that point, anything was better than nothing.
I was never a Kennedy supporter, but perhaps he was the right man for the job at that particular time.