I think a lot of the "conspiracy mindset" came about during the 60s. Prior to that, my sense is that the general public was far more trusting of the government and similar authorities (such as the Church). But perhaps due to the Cold War, Vietnam, the fight for civil rights, etc., suspicion and cynicism towards any form of authority started to grow. "Question authority" was the catchphrase which so few people utter nowadays.
But I also wonder about the opposite viewpoint. What motivates some people to step forward to challenge any criticism or theory alleging possible government wrongdoing? The government is certainly more than capable of defending itself, so there's no pressing need for any member of the general public to defend them or run interference against conspiracy theories as they often do. Some of them almost seem as obsessed as the conspiracy theorists themselves, as if their life depends on debunking this theory or that theory.
I realized this when I stated matter of factly that the JFK assassination was still a mystery, some people took me to task by essentially saying that "anyone who doesn't absolutely believe 100% that Oswald alone killed JFK is a paranoid conspiracy wingnut."
I will give conspiracy theorists some credit for at least allowing people to believe or not believe, but those who embrace the "official story" don't have the same level of tolerance, respect, or courtesy. Their general manner and attitude is that of ultra-orthodox believers who simply don't tolerate deviation or dissent.