President Eisenhower reflected later the public panic in the wake of Sputnik: "its light was blinding." The mantra that caught on was that the United States was "five years behind" the Soviet Union (false), that the Soviets could now threaten the American homeland with nuclear weapons (not quite yet, but soon), and that the whole strategic situation had radically changed (not quite yet). There was also a notion that Sputnik was a huge, secretive surprise (false — it was announced well in advance — the logo of the International Geophysical Year for which it coincided included a satellite on it). All of this added up to a lot of fear and concern that became a major form of political football and an opportunity for nearly everybody to interpret it in a way that advanced their own agendas.

From Eisenhower's point of view, the narrative ran away from any control. They had known the Soviets were working on this technology (so were the Americans), they had known that a satellite launch was imminent (the Americans had their own planned), they knew that this was going to be the trend in terms of both science and arms development. They also knew that early Soviet missiles would not yet be entering into production soon, that they were not yet reliable or accurate, and that the US still maintained a ridiculously overwhelming nuclear superiority over the Soviets (about 8:1 in 1957), as well as means of delivery (the USSR was ringed with military bases loaded with US nuclear bombers). (The Soviets did not really gain the ability to inflict reliably terrible damage against the American homeland until the early 1960s, and did not reach nuclear parity with the USA until the 1970s.)

And yet, public perception matters. The public panic fed calls from Congressmen for increased military funding. The already-powerful forces (the "Iron Triangle" in the language of sociologists) between Congress, the military, and American industry grew stronger. Eisenhower felt forced into making decisions he knew were bad ones, made from fear rather than reason. The amount of US missile deployments dramatically increased, even of weapons systems that were completely inadequate (like the Snark missile, most of which likely would not have gotten off the launchpad, or the Polaris warhead, which in its original form would not have detonated, or the Jupiter missile, which was inadequate and vulnerable as a second-strike weapon and thus could only be used preemptively but in a way that would give lots of warning), but increased the illusion of American security and safety.

These conditions are the motivating factor behind Eisenhower's later farewell address, in which he warned of the growing power of the Military-Industrial Complex, a power that even a beloved war-hero turned President could not adequately counter, in the fact of public fears.

There were other effects than just military-built-up, though. Education reform became the craze — it was felt that the US had "fallen behind" because of a lack of scientists and engineers, fueling a push for STEM fields that lasted until 1970, when the STEM job market crashed (because it turned out there was a finite need for STEM-trained people). In this sense, the aftermath of Sputnik reached into the lives of practically every American child, albeit in ways that they might find subtle and positive.

But on the whole, the US public was suddenly in place where it felt extremely vulnerable, and it did make a matter of doubt about whether the US system could adequately compete with a controlled economy and R&D system like the Soviet Union. Some of these fears were, even at the time, clearly overblown; in retrospect, many of them are based on very false assessments of what the USSR had done. (They basically skipped long-range bomber development and started working on long-range missile development earlier than the USA. The USA "caught up" and surpassed very quickly.)

For more information on Eisenhower's and the public's reactions, Z. Wang's In Sputnik's Shadow talks a lot about the various responses by the Eisenhower administration. On the education reforms post-Sputnik, the work of David Kaiser is excellent and authoritative. On Eisenhower's post-Sputnik missile deployments, and their problems, Eric Schlosser's Command and Control does a great job. The images I have linked to (the "containment" ring of bases and the missile vs. bomber numbers) were created by me for the purposes of lecturing on this topic (the source for the deployment types is the National Resources Defense Council, and the source on the bases is the declassified report History of the Custody and Deployment of Nuclear Weapons, July 1945 through September 1977).