Michel Van has hit it on the head: the Apollo Program was already dying even before Neil and Buzz stepped foot on the Moon.
Look at the funding levels by year here:
In real dollars, funding peaks in FY 1964-66, and then starts to tail off rapidly. Further Saturn V production was suspended in late 1968. By FY 1971 NASA was making do with less than half the budget it had had just five years before - and more drops were on the way. In short, Apollo shriveled up after Apollo 12 because it had been shriveling for a while - it just wasn't quite so obvious to outsiders.
So the important question is to ask why this happened, by way of figuring out how you can change it.
David Portree, who ran the BEYOND APOLLO blog at WIRED for a while,
zeroed in on the proximate cause: The Apollo 1 fire.
After the fire, NASA came under close scrutiny and was found wanting. Congress could not “punish” the agency by cutting the Apollo Program budget – to do so would have endangered achievement of President Kennedy’s geopolitical goal of a man on the moon by 1970, the goal for which the AS-204 astronauts had given their lives – but it could express its displeasure by cutting programs meant to give NASA a post-Apollo future. The agency’s FY 1968 appropriation was slashed to $4.59 billion, with AAP receiving only $122 million.
Of course, what Portree really doesn't go into is that, even without the fire, funding was already dropping, and it was always going to be a real push to keep Apollo going in some serious fashion no matter what. Apollo 1 probably cost a few Apollo Applications missions, but the full panoply of Apollo Applications plans (AES, ALSS, LESA on the Moon, lunar mapping missions, a Venus Manned Flyby, etc.) was never likely to survive intact, fire or no fire.
So can that be reversed or avoided? How can Apollo be kept alive and vigorous? We have had a fair number of threads (like
this one recently) asking some variant of this question, and the usual answer we come up with is, like Bahamut says up above, that the one real shot is a
more sustained and successful Soviet moon program - and as a corollary, that detente doesn't happen for a while longer. Either Korolev is given a green light a few years earlier, allowing the Soviets to beat the U.S. to the Moon, or they sustain the effort and attempt to one-up the Americans by establishing a lunar base. And this is not all that impossible, given that, even as delayed and disorganized as the Soviets were, they were
within weeks and even days of beating Apollo 8 to the punch with a cislunar flight (though a Moon landing before 1971-72 was not in their cards by that point).
Otherwise, short of Alien Space Bats, there's no real plausible point of departure. It was the Cold War that got Apollo off the ground in the first place, and only the Cold War can keep it going. There's no major domestic constituency for NASA otherwise. Indeed, polling shows that popular support for Apollo even in its heyday
was moderate at best, and declining even before Apollo 11. So you need an external driver, and the only one really possible is the Soviets, whose leadership doesn't have to worry about opinion polls or a rambunctious legislative process.
So if, somehow, you CAN get the Soviets to drive harder for the goal line (and set new goal lines beyond it), there are some timelines that you could look at. Here's three of the best recent ones:
1.
David Portree's continuing Apollo. David Portree himself looks at how Apollo Applications would have played out - Archibald set up a thread for that here. That timeline goes into the late 1970's.
2.
2001: A Space Time Odyssey (Version 2). Spacegeek, with help from Michel Van, constructs a timeline wherein Sergei Khrushchev, the General Secretary's engineer son, gets a job at the Soviet Experimental Design Bureau (OKB) in 1958, which later results in a more unified and aggressive Soviet space program, with all that entails for the Space Race. In progress, updated through 1968. Wiki page with dates and posts
here.
3.
Red Star: A Soviet Lunar Landing. Spacegeek and Bahamut-255 looked at what a successful Soviet Moonshot program - driven by Khrushchev giving Korolev a blank check in immediate response to Kennedy's speech in 1961 - would have looked like, and done for Apollo. I think that this may be the most satisfying attempt at answering your question, because it shows what a Space Race continuing well into the 1970's would have looked like on both sides. (Hint: Lots more boots on the Moon.)
Otherwise, if you want a more modest and realistic point of departure, our best ongoing timeline is
Eyes Turned Skywards by e of pi and Workable Goblin (with fantastic renderings by Nixonshead - see the
timeline wiki for an easier read-through, technical specs, and all the images). In this world, the decline in NASA funding is just baked into the pie, and the authors look at what a simple change in direction at NASA - Nixon appoints George Low rather than Tom Paine as Administrator, and Low opts to forego the Shuttle and stick instead with developing Saturn/Apollo hardware toward a more affordable low earth orbit space station program (Skylab, and a Skylab based long-term successor) for the 1970's and 80's. The good news is that this hardware family is much more suitable for an eventual lunar return in the 1990's, when the political will for it returns.
And since you mentioned Apollo 12, I thought I'd just leave you one visual teaser from the
Eyes Turned Skywards timeline: