Averting your eyes does exactly what it says it does: You willingly look away from your target.
Unless surprised, a creature can avert its eyes to avoid the saving
throw at the start of its turn. If the creature does so, it can't see
the medusa until the start of its next turn, when it can avert its
eyes again...
When you cannot see a target, you have disadvantage on attack rolls made against it (PHB p. 183) and many spells that require sight of a target will not work. If a spell indicates it affects a target that you can see, then it will not work if you avert your eyes. It's worth noting that averting one's eyes is not a common interaction in the game.
No, you cannot look at the floor and retain vision of her. If you avert your eyes, you cannot see her. If you choose not to avert your eyes, you might get turned to stone
Also, if you cannot see the medusa, she has advantage on attacks against you. PHB p. 195 :
When a creature can’t see you, you have advantage on attack rolls against it.
The same goes for monsters against you.
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.
Best Answer
There is a case to be made either way.
This is a great question, and it is going to come down to a DM ruling, as there is a compelling case to be made for either ruling. Unfortunately, I cannot give a definitive answer either way, but I can offer some arguments and let you decide. The first is a more strict rule-oriented approach, that is, let's just go by what is written without trying to make sense of it; and the other is a more "simulationist" approach, that is, what makes the most sense in the context of the narrative. The DM and the players should just work out how they want to rule on the Robe, and apply that ruling consistently over the course of the campaign.
Interpretation 1: You can still see in all directions, even while the robe is covered by another article of clothing.
There is a case to be made here based on the Robe's interaction with creatures that have abilities that trigger when they are seen. The medusa has an ability called Petrifying Gaze:
To avoid this, a creature can usually avert its eyes:
While wearing the Robe of Eyes, a creature is never considered to be averting their eyes:
If you can never avert your eyes from the Medusa, you can always see the medusa, even while wearing another article of clothing over the robe.
Interpretation 2: The eyes on the robe are doing the seeing, so they would see only the article of clothing that covers them.
Alternatively, we can make an argument from the spell description that the eyes are function as points of sight, and what you see is relative to their position on the robe. Usually magic items don't tell us how they work - magic be magic. But with the Robe, it seems to indicate how it works:
This seems to indicate that the position of the eyes matters, so covering the eyes with another robe would mean they see the inside of the robe only.