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.
On crystal.isles these are NOT tameable at the moment, so you will either have to spawn in a pair of magmasaurs and breed them for eggs or spawn rockdrake eggs, I would go for magmasaurs as they are breedable and rockdrakes aren't. Hope this was useful.
Trapping this creature does not work on official genesis. You will have to find a wall or overhang to grapple onto above the aggro basi and drop eggs down to them. This is the only method that has worked for me on genesis.
When it is underground it does use stamina which means it can not be underground permanently, it is extremely fast when it is underground and untargetable, and when underground it will be blocked by structures such as, walls, foundations, rocks, etc.
You Can Now Tame The Basilisk With a Fertilized Magmasaur Egg in The Volcanic biome on Genesis! Here’s A Tip if Your New To Basilisk Taming, I Majorly Recommend Placing Foundation’s on Your Trap Because They Will Borrow Underground if There’s No Foundation Placed and Lose There Taming Efficiency Also The Fertilized Magmasaur Egg Has To Be Dropped From The Players Inventory Not Your Tame! Good Luck.