I'm not sure why, if you want a wired ethernet connection downstairs, WiFi comes into the discussion.
Normally, with a computer remote from the router, connection would be wireless directly from the computer to the router without any cables, provided there was a good enough signal.
However, if you do want a wired ethernet connection without running a cable all the way, I'd suggest using Powerline adapters to link the router and computer via the electric wiring. You would plug one adapter into a socket beside the router and connect to it by ethernet cable and a second adapter beside the computer and run an ethernet to the computer.
Examples of Powerline adapters