It could be possible depending on the router you have. However it will not be easy. I think your best chance is try a custom firmware supported by your router, probably dd-wrt, check the documentation and try to follow some tutorials.
Another possibility is to get a router that creates two wifi networks. Some solutions to share the Internet connection works that way like the fonera. It will create a public network for everyone with a Fon account and a private network for you. It's not a very clean solution but is probably the easiest without expensive hardware.
A "No solution" that will just mitigate the problem is to change the parameters of the router momentarily so you have a different network when they are connected. There could be some conflicts and not be a practical solution if it has to be done many times or if your goal is preventing the other computers from accessing the same LAN and not just protect the pass phrase.
1What about VLANs? Would they work? I've seen modem/routers/APs with support for them. – AndrejaKo – 2011-01-18T17:45:17.087
which OS do you want to do the work? – akira – 2011-01-18T18:34:16.727
Why do you want them on a separate network? – maxelost – 2011-01-19T00:58:29.567