Whilst it's burrowed approach on the back of a drake or spino. Build 4 dino gates around it without the doors. Jump off your dino and it will come up from the ground and be stuck in the gates. Place eggs around the gates for an easy tame.
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
Get yourself a Rock Drake, Karkinos, Spinosaur or any other creature usually ignored by this creature, dismout to gain its attention and mount again; Drop the Rock Drake Egg from YOUR inventory and proceed to circle the Basilisk until it looses interest on you. Wait for the feeding interval to end, repeat.