Some conspiracy theories have a definite origin, a "Patient Zero" if you will; for example, the Phantom Time Hypothesis wasn't around until Heribert Illig came up with it
Other conspiracy theories are more organic; with the moon landings, some Americans were skeptical immediately.
A New York Times article from December 1969 noted people who claim the landings were "staged by Hollywood on a Nevada desert”. An opinion poll from 1970 put the number who "doubted the moon landings had taken place" at less than five percent, relatively normal for a conspiracy theory. (For comparison, a 2013 poll had 4% of people as believing in the existence of lizard people. I've heard this called The Lizardman Constant.) A 1970 pamphlet entitled "Did man land on the Moon?" is the first known printed source specifically aimed at moon-landing skepticism.
From Apollo 11: On the Moon: Look Magazine 1969 Special:
The almost perfectly executed odyssey of Apollo 11 seemed unreal to some. In McGehee, Ark., 81-year-old Mrs. Barbara Marion Hopkins Day was so unimpressed by the moon landing that she believed it to be a hoax, contrived for mere publicity. She slept through the moon walk on Sunday and did not turn on her television set during the entire lunar voyage. "I don't believe it," she said. "I don't believe they've ever been there."
A mere five years after the last landing of Apollo 17, NASA felt compelled to put out a document titled Did NASA Astronauts Really Land on the Moon? which discusses a book from the year before (We Never Went to the Moon) which counters a specific point of the author pointing out the lack of a visible crater; with the notes from NASA that there is some surface dust moved on landing, but the surface pushed the blast sideways. Additionally it notes that lunar soil is not "fluffy dust" but "moderately dense" more like wet soil so it is perfectly reasonable the moon landing did not make a crater.
Gallup polls at 1995 and 1999 put the "staged" claim at 6%, and the most recent polling I could find also places the number at ... 6%. In other words, there really hasn't been much change at all, even going back to 1969. There have been a number of "maintenance works" in keeping the theory up, probably the most notable being the Fox 2001 TV special Conspiracy Theory: Did We Land on the Moon? which (allegedly according to Fox) led to a 20% increase in skepticism in polling, but any such effect (assuming it happened at all, promotional ad copy should be treated with suspicion) has been transient.
While the US opinion polls have held relatively steady, the Russian ones have not. They originally had no thought to the moon landings being faked. They instead pretended they weren't interested in a landing in the first place (effectively changing any claims of Soviet objectives to be with automated Moon exploration, more detail here at the answer by /u/Dicranurus). However, after about 2000 or so, the number as shot up dramatically, with Newsweek reporting in 2020 that "about half" of Russians believe the moon landing was a hoax. (As far as exactly when or why this happened, the origin seems to unfortunately be past our 20-year window for this sub, but I should at least note it coincides with media becoming under much heavier state control.)