You are right to compare museums and libraries:
Some people question the function and responsibility of public museums to give all people chance to receive arts and knowledge, like public libraries do.
So why do you think our culture if OK charging (and paying) for museums and not for libraries? Is the notion of paying to use a library so abhorrent as to be taboo? Could we somehow transfer that to museums?
And, if so, what will plug the gap in the budget?