Obviously, the manned spaceflight program was extremely popular in the 60s and I can't imagine that the average American couldn't name at least one astronaut, but what what about before that, when the unmanned program was still getting together? Were launches televised? Were the failures covered as major stories?
Space exploration was a pretty hot topic in the 1950s, particularly amongst those who were interested in science of course. Many in the general public still considered space flight a pipe dream and did not give it much credence, but high circulation magazines like Popular Mechanics were discussing the idea of spaceflight and aiming for the Moon back then, and of course the Cold War between the USA and USSR was ramping up to the point that by the middle of the 1950s, everyone in America was well aware of the very real nuclear threat that was growing. What was needed was a delivery system to send those nukes halfway around the globe to hit the enemy, and it was steadily becoming appreciated by more and more people that ballistic missiles were going to be the delivery vehicle of choice for both sides.
1955 was a rather important year in the Space Race (it is reasonable to refer to 1955 as the publicly recognized "starting line" for the Space Race), because it was then that both the Soviets and Americans publicly announced their intentions to press forward and, within the next few years, send satellites into space by using ballistic missiles to launch them. That is when the general public really began to take notice, because if you could send a peaceful scientific satellite into space, people reasonably inferred that you could also damn well send a nuke downrange to hit another country with that same launch system.
The launch of Sputnik 1 on October 4th, 1957 came as a shock to most Americans, and really made the citizens of the USA very well aware, and very interested in, their own nation's space program efforts. Many were even panicked by the Soviet success of Sputnik 1 because they were not told it was coming or did not believe it was possible at that time, and that of course drew their attention to wondering how their own nation of America would respond, dramatically increasing the public's interest in the new "Space Race" that had suddenly been officially kicked off.
US civilian attempts at reaching space were, from that point on, HEAVILY publicized, because the American public was demanding a successful response to show that the Soviets did not own that high ground above, and the public wanted to see the American response that would prove their nation had viable boosters of their own that could serve as ICBMs as well as civilian-oriented launch vehicles. Due to its more open democratic ideals, America was forced to conduct their launches in full view of a critical public eye of course, whereas the Soviets, due to their very secretive political system, could keep any failures they suffered under wraps and largely hidden from their citizens. American launch attempts were absolutely televised and covered live over radio as well, and the public interest made sure any launches on the schedule became a major story long before the rocket motors were ignited.
On December 6, 1957, the first US attempt to launch a satellite resulted in an absolutely brutal propaganda failure when, in full view of the public and broadcast live on television, the US Vanguard TV-3 rocket exploded two seconds after liftoff, only getting to an altitude of four feet above the launch pad before collapsing into a ball of flames.
The failure of the Vanguard TV-3 was a MASSIVE international humiliation for America. Not just the American public, but the entire world knew about it, and also knew that the Soviets had already been successful with their Sputnik 1 and Sputnik 2 satellites. That put tremendous pressure on the USA, and the public was very much aware of the humiliation and were desperate to see a success to prove to the Soviets and the rest of the world that their military and technological strength could match the USSR.
When the US finally successfully responded with their Explorer 1 satellite launch on January 31st, 1958 using a Jupiter-C launcher, the American public was of course tuning in to watch very closely, and very proud of the success. The Soviets by that time already had two successful satellite launches under their belts (Sputnik 1 and Sputnik 2), so the success of Explorer 1 was a huge positive example for the US public to take pride in, because it proved they were actually capable of joining the race for space that Sputnik 1 had started.
So, in the USA, the public was very much interested in the unmanned space program after the success of Sputnik 1 kicked off the Space Race. The US failures were publicized and scrutinized heavily by the media, and the successes were milked for the positive propaganda narrative to the best of their ability as well, serving as a source of immense pride for the USA.