There is redundancy in the connector meaning there are multiple wires carrying 3.3v, so missing one is unlikely to make the system stop working, although I suppose it possibly could cause instability.
Your system would likely turn on without the +4 pin of the 24 pin plugged in at all even.
However, having a loose wire dangling around in the system is dangerous, it could touch something and cause a short, which at best would just turn the computer off, at worst would damage components.
I'd replace the power supply, or at the very least put electrical tape/heat shrink over the exposed wire