Actually the Basilisk only eats Rock Drake fertilized eggs. The wiki says that it should also eat other fertilized eggs, but currently it ignore them. Probably it's a bug, for now feed it Rock Drake eggs
Seems like raising a rock drake 1st will help you tame these. 1st raise a drake. 2nd collect a bunch of rock drake eggs. 3rd find a basilisk and while riding the cloaked drake drop the eggs one by one and it will eat them.