drivers? Well realistically no, most of the system in windows will automatically configure your wired connection once you make it through the ethernet in the back of your pc(for the wired connection).
now for the wireless pc's they need a wireless card in order to take advantage of the wireless system as you probably already know. Wireless cards come in the form of PCI or USB devices that basically act as an antenna for the wireless signal. If you have a total of two routers, the wireless router that has the ethernet connection from your modem will broadcast the wireless signal.
Now unless the signal is weak, you'll have to move your wireless router downstairs in the basement but make sure you have a cable modem there to provide a signal to the wireless router.
You don't need a pc connected to the wireless router, you can just sit the cable modem next to the coxal output, run the main ethernet connection into the 'internet' part of the router and then plug them both in for power and the signal starts.
So that should, pressumably take care of one problem. the computers downstairs on the wireless signal.
No.2 would be the wired router upstairs(elsewhere in the house) that will involve the use of the 4 ports on the back of the wired router. It's basically the same connection as the wireless router, you plug in the internet to it and run one ethernet cable to the other computers. The nice thing about wired routers is they act as a fine switch so you don't lose any connection speed once you plug in more then one computer.
finally, the drivers, as I mentioned before windows is basically automatic and up to date, if there is an ethernet connection on your motherboard and on the back of your computer it's going to take advantage of the connecetion as soon as you plug in the ethernet cable to the port.
Unless you have some additional drivers or you haven't updated the software with the wireless computers(like drivers for each wireless adapter) then you should be set. Keep in mind that the wired connection won't be encrypted like the wireless connection should be, (WEP or wireless encryption protocol), this keeps people from connecting to your wireless signal and possibly infecting your pc or leeching your internet for free!
To set up a WEP there comes some instructions with the linksys wireless router that run you through configuring and setting up everything. I believe you need to 'open ip' address on the router then configure it through the browser window.
hope this all helps.