Milton Sheaf is the current Head Librarian and is charged by the crown with the upkeep of the library. He was one of the few librarians of Stormwind to survive the Orcish Wars, and spends much of his time restoring works that were recovered from the ruins of Old Stormwind.
The library is selectively open to those who gain permission from one of the royal librarians. One can go to one of the many public libraries throughout the kingdom and request an item from the Royal Library if they can not obtain permission to access it themselves.
Adapted from Wowpedia