Very briefly, say it isn't so:
Hollywood and our office of civil defense fed this fear. Montages in Sputnik Mania attest to the proliferation of apocalypse films, and clips from government advertisements include messages recommending that individuals build shelters and build them right [away].
I thought we never propagandized our own... never... Smith-Mundt made sure of that, right? Didn't it?
On a serious note, read from the Global Media Project:
When the Soviet Union launched Sputnik, “the first man made object ever to leave the atmosphere and successfully orbit the earth” on October 4th 1954, America reacted with fervor (Sputknikmania.com). It was the height of the cold war and this bold display of Soviet strength struck terror in the hearts of political and military strategists who saw in the rocket “an intercontinental ballistic missile that could potentially carry a nuclear bomb.” On the Monday following Sputnik’s launch, “political and military leaders appeared in print, on the radio and on TV, telling [the American people] that Sputnik was a threat to [their] security [and] that it was launched as an aggressive attack.” Sputnik, they said, was “the first shot in a cold war that could quickly become very hot” (Sputnik Mania).
Missing is the impact of the "CNN Effect" (yes, CNN didn't exist and yes, I agree the "Effect" in our time is less than advertised, but you get the point...): the leadership initially felt the launch was a non-issue. It was only after pressure from the media and Congress did the propagandizing begin.
(h/t Tim at Ubiwar)