How do I physically prevent people from unplugging an Ethernet cable?

I think this might help you http://www.rjlockdown.com/patchcordpage.html

I was looking in buying something like this a while a go, but I never did, but it seems to be what you are looking for

If you are looking for a cheap and easy way to do this, just use zip ties. You can put one near the end of the ethernet cable, one somewhere on the back of the fixed PC (like to an open vent or empty card slot), and use another to connect the two. It wouldn't keep the cables from being unplugged, but it would prevent people from moving the cable more than a few inches from the PC, rendering it useless to use on their laptop.

Don't fight; be polite.

Instead of fighting people who want to use the internet, make it easier for them to use the internet without interfering with the desktop systems in any way.

The first step to keeping people from doing things you don't want them to do is to ask them nicely.

1. Put a sign on the back of desktop systems asking people in need of ethernet cables to request one from the lab attendant. _{you can still ziptie the cables to the desktop, but that won't stop anyone with a penknife or pair of scissors.}
2. Get some cable. I recommend buying a large spool of ethernet cable, a set of crimpers, and some plugs. I recommend this over buying pre-made cables as it tends to be much cheaper, and you'd be able to make cables in whatever size you need. Most people shouldn't need more than a few feet of cable if the ports are close to desks. If people steal their cable, it'd be much cheaper to replace. Instructions on how to crimp an ethernet cable can be found all over the internet. It's remarkably easy to do.
3. Attach a sticker to each cable that asks the borrower to return the cable to the attendant when they're finished.
4. Give cables to people freely when requested. This is the most important part. They need a cable. If you meet their needs, they'll no longer have a reason to steal cables from other machines. _{You could set up a bartering system where they give you something (like a shoe, bag, or ID) and they get it back when they return their cable. I don't like this solution as it makes it easy for the lab attendant to get in trouble when someone's stuff gets stolen if they happen to go to the bathroom.}
5. Ask people to donate spare cables to support the lab. If you're giving away free cables, I'm sure there are others who have some lying around from their old router or computer.