At the risk of oversimplifying things, there were to some extent, but the space race had so many other things tied to it (notably the whole Cold War mentality in general) that it could never have been a factor in the development of the program and any vocal critics would not have found much traction.

Part of it was that everything was put together so deliberately; Mercury led to Gemini led to Apollo. It was well planned and very easy to point to that plan. Each successive mission helped prove that one thing or another was possible or provided a means of testing the potential of success for the next one. Easy example - Apollo 10 had the lander and they took it down almost to the point of landing, and then went back up to the capsule, just to be sure that everything up to that point worked correctly for Apollo 11. Just to be sure that the astronauts didn't get greedy, the engineers didn't give the lander enough fuel to launch from the surface, only enough to do the letter of the mission and no more. It's well known that 10 was something of a rehearsal for 11, but truly the entire program was like that dating back to the development of Mercury.

The shuttles were both wonderful and terrible because they made space commonplace and something that we could take for granted. Focus was lost for the public at large - although the shuttles were truly the next step in the progression that dates back to the earliest days of NASA, for they and the ISS enable us to explore what long-term habitation in space does to humans, in preparation for future endeavors. That's far less "sexy" than going to the moon, or taking the first space walk, especially when there isn't the air of nationalism and urgency that the Cold War-era Space Race brought to the table.

You also have to put into perspective just how much Sputnik scared people. The Eisenhower Administration did see a lot of the Sputnik spectacle as a stunt - though they had very secretive plans for a satellite program - but the reaction to Sputnik created an enormous amount of pressure on Washington and Kennedy's portrayal of the Eisenhower administration's inaction was a factor in his victory over Nixon. It was a big confidence hit to see and hear this Russian satellite after the strides that American society had taken in the wake of WWII. It's hard to say that Kennedy had a mandate given how close the election was, but nonetheless many were with him when he made his 1961 pronouncement about making it to the moon by the end of the decade - Especially given that Russia put the first man in space just a month prior.

I'd be remiss, of course, if I didn't not the economic climate these days as well.

Sources: Documentary "When We Left Earth"; Nova "Sputnik's Impact on America"; Space.com "How Sputnik Changed the World 55 Years Ago Today"

Edit: As has been pointed out by /u/WalterSkinnerFBI, "Spaceman" was actually released years after Ochs died, and so it wouldn't have had a public impact at the time. I'll leave my answer because it still shows that there were critics of the program, but I wanted to point out that it has less weight than I initially thought.

I answered a similar question here last month. I apologize that it's not the most helpful answer possible, but it does answer your question. Yes, there were those who protested the cost of the space program, though the example I used was a leftist comparing the costs with the needs of the poor, rather than members of government objecting to federal spending.

I don't know how widespread the criticism would have been, but it did exist with some visibility in the American left. One of the more popular and influential protest singers of the 1960s and '70s, Phil Ochs, referenced the space program in two of his songs.

From the song "Spaceman" [song]* [lyrics]

Way high, so high:
Travelin' fast and free.
Spaceman, look down:
Tell me what you see.
Can you see the hunger there
Strike without a sound?
Can you see the food you burn
As you circle round?

From the song "Tape from California" [song] [lyrics]

Half the world is crazy and the other half is scared
Maddonas do the minuet for the naked millionaires
The anarchists are rising while we're racing for the moon
It doesn't take a seer to see that the scene is coming soon

Ochs was best known for his songs against the Vietnam War, including "I Ain't Marching Anymore," "Draft Dodger Rag," and "The War is Over." He had professional and personal relationships with other famous singers such as Bob Dylan, Pete Seeger, John Lennon, and Victor Jara (the latter's torture and murder in Chile would contribute to Phil's spiraling depression that eventually ended his life).

*The old link was dead, so I changed it.