Should frum children have a public library card? No, they should not. There are many books that are extremely inappropriate for children, even in the children’s section. I learned that lesson many years ago, picking up a seemingly innocent book in the teenage section, and read of subjects that I should not have read about, and wish I never did.
There are Jewish Libraries ask around in more densely jewish neighborhoods. Ask around and you will find them.
While more of the recent Jewish Adult novels coming out are dealing with peoples’s issues or problems. I think and feel if someone knows how to write books, we can use more authors who will write with the proper hashgachas that we as Jews need and want to imbue in our children.
If one lives of town, and there is limited access, to kosher reading material, I think Torah U’Mesorah has a book list of ok reading material that can be used to pick up appropriate titles. But to let a child into a library unsupervised into a public library, with access to any all types of shmutz is inadvisable and asking for trouble. It is like asking someone to to go to a candy store and pick out from candies that does not have a kosher mark on it and let them guess which ones are kosher. Reading inappropriate material is just as bad as eating trief IMHO. They both will effect someone spiritually, and can have negative consequences. As a parent, are you willing to take that chance?? While we can help what we see on the outside world, we as parents are definitely responsible for what we are bringing into our homes. Same way as we have the responiblity to bring in only kosher foods.