My argument against most conspiracy theories is simply: that many people can't keep a secret. Imagine how much a story like "We faked the moon landing, here's proof" would go for. Especially pre-internet, when everyone got their news from the actual news.

Also, if the US lied about going to the moon wouldn't the Soviets have also denied that it happened, instead of very matter of factly reporting on the success of Apollo 11?