Seems a little bit optimistic; could you imagine if the Cold War's space race had the additional prize of an archive of advanced alien technology?
Oh, I concede that there is no guarantee that an organization or country advanced enough to launch a mission to the Moon would be responsible custodians of the information (alternate timelines where Germany or the Soviet Union land men on the Moon come to mind). But they might be less inclined to destroy it outright instead of seeing what they can use for themselves.
The best we can hope for is to make data available to the future; what they do with it is ultimately out of our hands.
Edit: Just noticed your last remark. What does responsibility have to do with being able to access an archive of human knowledge and culture?
None of this means that multiple, redundant archives cannot be created. I just think that the Moon offers a unique level of protection from those would would destroy or otherwise degrade such a collection.
I'm of the mind that burying such records in a highly-conspicuous site on the Moon might be a better solution. Much less risk to geological processes or tomb raiders; if you can get that far from Earth then you might be responsible enough to deal with the archive.