Obviously you would have to have some other POWERED piece of kit (ADSL modem router, which you presumably have upstairs at the moment unless you are just using a USB ADSL modem now) located by your downstairs phone point and then route the Cat5 cable upstairs.
The Cat5 cable is thicker (more diffiult to route) and there will be a limit on the length of the run. SO it won't necessarily help.
Alternatve might be to have a wireless router downstairs and avoid the cabling issue.
Having said that, the telephone cable running upstairs may not be the problem (esp. if it is wired in correctly).