Title pretty much explains it.
I know about the lunar reflectors but other than that I can't really come up with any really strong arguments to prove that the moon landings did occur. I believe that they did but I feel like certain conspiracy theorists would destroy my points fairly easily.
Remind them that the US was in the middle of the Cold War with the Soviet Union, and a huge amount of prestige was at stake. Given the consistent expertise the Soviets displayed in both their space and espionage programs, wouldn't the Soviets discover that the US had faked some portion of their progress? Wouldn't they take the opportunity to expose the US deception to the world? If the Soviets believed the US claims, what standing do the skeptics to believe otherwise?
It's hard to argue with them because they're believing in a conspiracy that has faked everything related to the moonlandings.
For instance, you could show them these pictures of things (instruments, footprints & tracks) left on the moon, but they'll just say these too are faked:
(I picked the first picture I could find of every mission, there likely are better resolution images out there.)
Taking your question as face value, and not as a historical one, arguing is actually not the best strategy, if your goal is to change their mind that is. Even using rock solid evidence, it's been proven to reinforce strongly held beliefs due to cognitive dissonance, and particularly if they go against the commonly held opinion. Indeed, people who hold such beliefs have had them challenged already, and have had plenty of occasions to rationalize them.
I can't seem to find the paper at the moment, but as far as I know the only proven way to change people's position appears to be to mock their beliefs, unfortunately.
