If Nixon scuttles the Apollo program in 1969, he loses 1972 by a landslide.
The Democrats (and probably most Republicans) would have crucified him, and rightly so. An overwhelming majority of the American people were solidly behind the idea of getting Americans to the Moon first. No matter what you think of his politics, he was first and foremost concerned about his image, especially if the Russkies beat us.
I have to believe he knew a decision of that magnitude -- no matter how he played it off as "tax savings" -- would have been a huge blow to the nation's psyche, as well as a massive hit to the military-industral complex that was involved in producing GNP and paying tens of thousands of people to do it. Those corporations which lobby really hard in Congress to keep our military No. 1 in the world.
It's impossible to explain how Moon-crazy this nation was in the mid-60s. Tang. Space Food Sticks. Walter Cronkite live from the Cape or Mission Control in Houston.
Almost everyone knew someone connected to the space program or had a friend of a friend at Boeing or McDonnell-Douglas or Rocketdyne or one of the hundreds of contractors working on everything from hardware to software to sewing patches on spacesuits. It wasn't just in Titusville or Houston or Huntsville. People were employed all over the country as part of the push to the Moon.
The hippy-dippies didn't vote, as McGovern found out. But Ma and Pa America did, and they were solidly pro-American, anti-Soviet. If that meant winning the Cold War by sticking a flag on a cold, dark, airless satellite orbiting the Earth, so be it.
Nixon would have had more success getting out of Viet Nam at that point than shutting down NASA.