Yeah, I think I'm messing with your guy's brains by asking for a mundane NASA, because replies keep coming in about the grand space program AH 99% of the time loves to create.
Well, not surprising given the interbreeding between science fiction and alternate history. Anyways, I think there are really three kinds of mundane NASA, two that diverge about 1970 and one that diverges about 1957, at least among the ones that have NASA in any recognizable form. There are of course a lot of timelines where, say, nuclear war in 1962 rather reduces any appetite for space exploration, or where both the Nazis and the Soviets were so devastated by World War II that the Cold War was totally averted, or where pre-WWII or even pre-WWI power dynamics keep going long enough that there's something like a multipolar space race, or where in some other way NASA was destroyed or rendered unrecognizable, but those aren't as interesting.
The two that diverge about 1970 are basically OTL-type NASA and ETS-type NASA. That is, on the one hand you have a NASA that develops Shuttle, and on the other one that develops stations.
How they do it gives birth to a range of interesting timelines, but is in principle irrelevant to the classification of timelines. In either case, NASA gets locked into working on whatever they choose for a long while and quite likely will allow important skills and capabilities to degrade, if only for the lack of funding most likely available in the 1970s. You can read our timeline or an actual history book to get some feeling about how these turn out.
The third is closer to Polish Eagle's Scenario 2, one where the space race and Kennedy's pledge to go to the Moon are averted altogether (mostly by the US being a bit luckier and the Soviets a bit less early on). I don't think it was as avoidable as he thinks by 1961, and I am less pessimistic about the likelihood of maintained orbital presence than he is (if only because the Soviets are not likely to give up on military space stations and the like until later, and that will surely convince Congress to respond), but in general I think that's a good outline. Not as much interest in human spaceflight, a token maintained presence in a small space station for pure prestige reasons but little real interest in exploration. Possibly more interest in robotic exploration, or possibly less; in either case, there would be less motivation to support possible HSF targets (ie., Mars). Quite possibly science fiction would be even more influenced by the various -punk strains than it actually is, with more focus on future Earths without any real spaceflight; alternatively, given that science fiction actually developed in an environment where space flight was impossible, science fiction might go really hard for expansive space futures. Or both; certainly the classical authors will be pushing for the latter by the 1970s and 1980s, and the new generations will probably end up split.
Of course, the best alternate mundane NASA is one where spaceflight developed gradually out of sounding rockets, and NACA got put in charge of US scientific satellite programs--think Vanguard, Explorer, HEAO, etc., not Voyager or Mariner and certainly not Mercury or Apollo. It's basically a totally boring science agency with the public profile of NOAA or the NSF.