How's it possible that Dumbledore was not aware of the basilisk?

Question

How's it that Prof. Dumbledore was not aware of the existence of the creature basilisk, while Hermione learned about it in a book?

Answer 1

There is no reason to think that Dumbledore was not aware of the existence of the creature basilisk. There is an explicit entry on the basilisk in Fantastic Beasts and Where to Find Them, which is a first year textbook. Dumbledore himself wrote a foreword to Fantastic Beasts in which he notes its ubiquity:

No wizarding household is complete without a copy of Fantastic Beasts, well thumbed by the generations who have riffled its pages in search of the best way to rid the lawn of Horklumps, interpret the mournful cries of the Augurey, or cure their pet Puffskein of drinking out of the toilet.

Additionally, when Harry returns from the Chamber of Secrets he explains the whole story to a room filled with Dumbledore, McGonagall, and Mr. & Mrs. Weasley, and there is no indication that he needed to explain to anyone what a basilisk is.

Knowing of the existence of such a creature was not the issue. The issue was that no one had ever actually proved that the Chamber of Secrets even existed, let alone that it was inhabited by a basilisk. The reason that Hermione was able to figure out that the basilisk was the monster of the Chamber of Secrets was that she was privy to information that Dumbledore did not have — namely, that Harry was hearing voices inside the walls that no one else could here. Knowing that Harry was a Parselmouth she was able to figure out that the voices must have been serpentine.

Had Harry confided in Dumbledore about the voices he was hearing, Dumbledore might have figured it out even faster than Hermione.