Cheapest option is put a wireless card in each machine and use the Internet Connection Sharing (ICS) built into XP. You can pick up cards on ebay for about £20 each + p&p, so the cost will be about £50. (I'm assuming your new machine is running XP).
That option requires both machines to be on for the kids to use the internet. You will have limited or no parental control.
Option two is to use a wireless router with your machine connected to it by cable and a card in the kids. You can pick up a cheap wireless router on ebay for about £40. Add the additional network cable and your are looking at about £70.
That assumes that your modem has a network connector to plug into the router. If it's a USB modem then you'll need to replace that as well. A cheap combined router/modem is about £65, making the cost about £90.
The advantage of the router is that it will provide a DHCP and NDS service, probably have a firewall, and will allow you to control internet access in a variety of ways. As long as you ensure only you have the admin password.....