More can be said, but these previous answers might be of interest

How did people and policy-makers react to Eisenhower's "military industrial complex" speech? It seems like it didn't have much of an effect. Why is that? written by u/restricteddata touches on this topic briefly

How do we reconcile Eisenhower’s apparent fear of the Military-Industrial complex and his role in leading us into anti-communist wars? written by u/DBHT14

Adding to what u/restricteddata wrote, Eisenhower farewell address was given on 17 January of 1961.

The Sputnik had launched on 4 October 1957, and the perceived loss shaped the rest of his term. Yuri Gagarin, the first human in space, had not launched yet. That would come on 12 April, after the address.

Going back to 1957, before Sputnik:

The question of airspace was critical during Eisenhower's presidency. Both sides of the Cold War were keenly interested in knowing how many nuclear missiles the other side had. Of course the US wanted to be able to fly high-altitude spy planes over the Soviet Union... and did NOT want high-altitude Soviet spy planes flown over the US. So, the question was: how high does a nation's airspace extend? Does it extend to Earth orbit? (If so, making Earth orbit - nearly any spaceflight- against international law.)

Both Sputnik and Explorer were intended as part of the International Geophysical Year for international cooperation in science. As with the later NASA budget, the scientific achievements had dual purposes: Civilian, yes, but also Military.

For political reasons, the US wanted to launch the scientific satellite on a rocket viewed as scientific: the Naval Research Laboratory's Project Vanguard, and not the US Army's Atlas rockets being developed by Werner von Braun.

One historical question, then, is: did Eisenhower intentionally let the Soviets launch first? The US had the capability, and von Braun was raring to go.

Had a US satellite orbited first, and the Soviet Union objected? Then the international space law would be muddied for years.

By allowing Sputnik to launch first, and NOT (politically, internationally) protesting its flight over the US, the default space law allowed orbits. The US followed with Explorer 1 on 1 February 1958.

What Eisenhower may not have anticipated, was the public panic that ensued. This kick-started an education uproar, leading to schools pushing STEM (Science, Technology, Engineering, and Mathematics) subjects and funding students pursuing STEM degrees.

The same concerns led to the creation of NASA as a Civilian government agency in 1958.

Spy satellites took a little longer to become operational after the first satellites were launched. Film was first recovered in 1960 At first, the film was dropped from orbit and caught by aircraft for processing.

Meanwhile, the U-2 program continued... and became public knowledge in 1960.

So Eisenhower's farewell address spoke to the public reactions to Sputnik, the beginning of NASA, the continuing Cold War... possibly even to the misinformation affecting political decisions. Eisenhower knew that Kennedy's "missile gap" claim was false. But he couldn't reveal what he knew, or how he knew it.

References: Burrows, William E. "This New Ocean: The Story of the first Space Age." 1999.

McDougall, Walter A. "The Heavens and the Earth: A PoliticalHistory of the Space Age." 1985.

If you want to know more, I haven't actually read: Burrows, William E. "Deep Black: Space Espionage and National Security." 1988.