Approach with Drake so it doesn't aggro, surround in Stone Dinogateways to trap it. Kill a nearby creature and leave its corpse, aggro Basilisk by hopping off and remounting Drake, run away to de-aggro. Basi should aggro onto corpse we left. Toss eggs and tame, kill and leave a new corpse before the first expires, rinse and repeat as needed.
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.
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
3 min 30 second feeding interval it seems. I have better taming effectiveness by keeping it aggroed while running it in circles. Lvl 20 only took 2 eggs. Lvl 85 took 4. Drop 1 egg. Aggro. Let it feed. Start stopwatch. When ready to feed it rears up n screams but does not spit