Wick3 said: "who archives it? The NY public library? Does the public have access to these archives after a number of years? Do they film (for archival purposes) all Broadway shows?"
I think they record pretty much everything, including Encores! and such. They are available per request, and you need to provide a reason why you want to view a certain show, e.g. musical theater students or journalists doing research. However, when I went recently I found they were pretty chill, at least on that occasion. I filled out the paperwork, and nobody asked any questions and I didn't need to provide proof for my research reason. You do need to have a library card (although nobody checked in my case)
However, you can only watch them at the Library in their TV room, on pretty old monitors (ok quality) and with headphones.
If you go on the library website you can read all the rules and regulations in detail also check the catalogue. It's pretty amazing. I would like to lock myself in their for a week and just watch all the old shows!