There should not be public libraries, if by public one means government funded/operated. There is no sound reason for government to be involved in libraries from a minarchist point of view. Especially nowadays when access to books is widespread via the internet and when one can acquire books electronically on devices such as Kindle for little cost. I even find lots of the old theological books, as well as old hymn books with theologically sound lyrics, such as by Joseph Irons and Isaac Watts, that I like, for free on the internet.
So, so many of these cultural issues would be WAY less divisive in the United States if government, on all levels, stayed out of them.