If you fail your first save against the basilisk's gaze, the second happens regardless of whether you can still see the basilisk.
Your reading of this rule is correct.
Now on to weightier matters - how do you make this encounter feel fair?
Give them warning signs.
The drawback that the robe of eyes has is quite deliberate and you shouldn't make it trivial to circumvent. What you need to do in this circumstance is offer the player the information necessary so that, if they're paying attention, they know they should probably remove the robe of eyes before they ever see a basilisk.
Luckily this is very easy to write, since monsters that petrify leave obvious signs of their handiwork - petrified people all over the shop! Just make sure that you describe the strange statues they're seeing, and if you want to give your player a particular hint, you could even specifically say something like:
"Towards the mouth of the cave, you see a handful of strange statues, like a band of adventurers getting ready to head in. In fact, [Player], with the robe of eyes you realise you're actually surrounded by strange statues, some so overgrown with moss and vegetation that you didn't recognise the humanoid shape at first - and all of them are looking at the mouth of the cave, whether their bodies are facing toward it or looking back over their shoulders..."
This lets you telegraph that there are petrifying monsters coming up - it reminds the player that they are wearing the robe of eyes - and it gives the very strong clue that it's a gaze attack, not just some grumpy wizard, that's turning things to stone, because all the statues are specifically looking at the same thing. In such an instance, if the player continues wearing the robe of eyes, and strides confidently into the encounter with the basilisks, then they should probably be able to recognise afterwards that they should have known better!
If you need to, you can start foreshadowing even earlier than this. Maybe your party isn't likely to know about basilisks, so they wouldn't (without metagaming) be able to put two and two together at the mouth of the cave - but then it so happens that a bard at the tavern they stayed at the night before was regaling their audience with the legend of the Medusa, so they're primed to know about creatures with a gaze that can turn you to stone. (I'm especially fond of this idea because then you get to subvert their expectations that they're about to stumble across a medusa so they can still be somewhat surprised without their deductions being useless, plus it's funny.)
Let them see the basilisk first.
The Petrifying Gaze only functions if both the basilisk and the target can see each other. If the basilisk cannot see the character, they are not affected by its gaze! If they're aware they need to protect against a gaze attack but haven't yet removed the robe, contriving events such that they see the basilisk before it sees them - perhaps when they first come across the creature it is distracted by something else, consuming a meal, or as a reward for successful stealth if they are attempting to do so - allows the character an opportunity to remove the robe before they can suffer any ill effects.
Make sure they can recover.
If you don't want to deal with a particular character being out of action for a long while if they should happen to be petrified, make sure that the resources are available to cure them afterwards. A scroll or wand (with few charges) of Greater Restoration might just be luckily found in the possessions of a less fortunate adventurer or in the basilisk's lair, if the party aren't able to reverse the condition with their own magic or resources.
It's not actually that likely they'll suffer petrification, even with the robe of eyes.
The DC of the basilisk's gaze is only 12, so your player probably has at least 50:50 odds of passing the save - and they have to fail twice in order to be fully petrified. Though that's within the realms of possibility, so it's still a threat, they'd have to be unlucky to do so, and a reasonable player should be able to recognise that (though it could be understandably frustrating!) Certainly it would be very over the top to describe that encounter as railroading or forcing them to fail.
