The J-lem public library system has a small selection. However, if you wouldn't send your kids to a public library in the states, you shoudn't send them there, either. You can go yourself and pick things out, but you may have to work harder figuring out what is written for whom. (You'll see books about Shabbos with non-tznius pictures, for instance.)
When I opened an account, I needed to give them an Israeli check as a deposit.
The biggest branch is located on Betzalel right near where it hits Ben Yehuda.