Chamber of secrets, anyone?

how do you train it

Anyone good a good idea how to return a petriefied player character to his normal form, if the party doesn't have a greater restoration available and also no NPC that is able to cast it? Additionally he is trapped in an cave, that is only reachable by water.

You could have them brew an antidote from the blood of a basilisk. Set some DC on the success rate. If the one trying to brew the antidote messes-up  horribly, they petrify as well. If they use the blood of the basilisk who petrified the player, give them a bonus on the DC. Could end up in a nice hunting session.

Thanks for the nice idea. I will try to propose this idea to my players, even if the petrified one has created a new character already. I am curious, how they will handle the situation.

Step by step guide to taming a basilisk. If one option is unavailable to you due to your level, move on to the next one. Keep in mind that a CR monstosity isn't likely to make a safe pet, or a good one, but hell let's give it a good go!

1. It is unaligned, but also a monstrosity, so the best system would be to cast Dominate Monster on it and treat it very kindly. Have it gently interact with you while you pamper it and speak with it. Try giving it it a little more autonomy as time goes on if it's going well. Make sure that it's immediate needs are met, especially food. Continue this until the creature begins to consider you a source for food and comfort rather than as comfort food.
2. If you're not high level enough for an 8th level spell, try charm monster, which is almost as good in this situation. I suggest following up step one with this once you're out of 8th level spell slots. Remember that you can upcast this spell to use it as many times as necessary to train the creature. I recommend taking at least one full day to train it in this way, but preferably a week. If you're level 11, you can reasonably do this for six hours a day, so keep at it.
3. When all else fails, cast Command on the sweety, especially if it starts getting bitey. If the creature is able to attack you in any way, than the charmed condition from steps 1 and 2 will have failed and the spell is done anyway, so no use crying over spilt spell slots and concentration. Cast command while you let yourself recast the spells from step 1 or 2 next turn. It'll show the creature that you are boss, and reinforce behavioral changes. This is basically the equivalent of a rolled up newspaper or spray water bottle.

If you've done all of this successfully and your DM is cool, they should allow you an animal companion Basilisk. Just keep in mind that it's still a dangerous creature, it's a medium creature, it's a monstrosity that turns things t stone with a gaze, it needs to eat at least as much as you do. While training it, take some extra time to teach it eyeball manners/to wear blinders (like a horse).

I'm assuming your group has handled this by now, so this might no longer be meaningful. The stat block does say that whenever it consumes the creatures it turns to stone that they turn back into their organic form. If one of your players reached its arm down the throat of a freshly dead basilisk, maybe they would get coated in whatever chemicals start the transformation process. 

That is not a snake.

neither does it have chicken parts either!! Lol but let's be real here, if a snake had 8 terrifying legs and spikes all along its back, wouldn't it just look like a basilisk? 

If you want to tame a basilisk, you can cast Awaken on it and treat it well before the charm ends, and you have a sentient basilisk companion since it has under an intelligence of 3

The gland of the basilisk could be used to cure the petrify fi they boil it or some other method, a bad check to get the gland could result in them getting pretified. 

Step 1: make a new character

Is the creature effected by it's own gaze or is it immune to turning to stone?

1

Your campaign has already passed beyond that point I assume, but if you feed him a piece of basilisk gizzard it will cure him. The lore states that a basilisk gizzard turns the stone statues into flesh which it can digest. This means that if you kill one and grind up its gizzard into a raw patty, and then feed it to the petrification victim, you should be able to cure them.

yes chamber of secrets

Yeah

Can these guys dig it do they find already existing caves

how fast is it?

I think you're forgetting, the victim has been petrified.  Essentially a statue.  You can't feed a statue.   It's more effective/realistic to create a potion or a balm that can be applied to the creature topically.