The first thing you need to check is if your cable modem is capable of connecting by ethernet.  If not, you will need a new cable modem, You first then need a router to share the internet connection.  If it is convenient to run an ethernet cable to the computer in the basement, a regular wired router is necessary.  You would then connect the cable modem to the routers WAN port (also sometimes called Internet port), by an ethernet cable and connect each of the computers to one of the 4 LAN ports on the router by etherent.   Of course this requires that both computers have ethernet network adapters.

 

If it is inconvenient to run an ethernet cable to the basement computer, you could get a wireless router.  The upstairs computer would still connect to the router's LAN port by ethernet, but the basement computer would require a wireless network adapter to connect to the router wirelessly.   You two options for a wireless network adapter asuming the basement comuter is a desktop computer.  If it has a free PCI slot, the PCI wireless network adapter would work.  Otherwise you could get a USB wireless network adpater that would plug into a USB port on the downstairs computer.   A wireless router will connection of up to 4 computers by wired etherent and up to 250 more computers by wireless.

 

Steve