is to buy an ADSL modem/router combination, and to fit a wireless network adapter to the second PC. The upstairs computer can be connected to the router by an Ethernet cable (that computer must have an ordinary network card installed, but pretty well all computers do).
Once the router is setup - an easy task - the second computer will be able to use the broadband connection at any time, without the need for the upstairs machine to be turned on. The router will establish the connection automatically each time it's switched on. It's not a good idea to leave a router connected permanently, so turn it off at night.
If you need specific advice about which router or network adapter to buy, post back, and one of us will make recommendations.