In terms of this, I think it has nothing to do with your eye nerves or something else. I think it is just a matter of your habit. Think of this, you always do everything with your eyes open. No matter when you work, you play or even sometimes when you just sit here but do nothing. You feel safe cuz you can see everything and you know that you will not fall down suddenly or something else. However, when you close your eyes, your sense of safety will be gone and you will be trapped in fear. You don' t even dare to walk for a small step. Naturally, you will lose your balance, lol. This is how I understand.
I think closing our eyes will affect our balance if we are not blind guys. But for those born with blindness, they have learned to keep balance by other senses. Usually all kinds of our feelings work together to keep the balance of our body, but when we lose one of them, our body will learn to keep balance by the rest of the senses. When we have our eyes open, we can adjust our body against a fixed position to correct our movements, but we often do this unconsciously. When you close our eyes, we need to adapt to the condition for a long time before we can correct your position with eyes closed, so we may lose the balance.