It is likely what others have said above. Go into your BIOS (the manual for the PC or motherboard should say how to do this) and disable WOL (Wake-On-Lan) if you can. It may or may not work.
Both of my PCs here have dual LAN on them. One port is active all the time and the other one is only active when the PC is on. If you have dual LAN ports, try the other one.
I have a UPS with a power monitor on it. When my LAN is active but the computer is off, there's a difference of 1-2 watts. You would have a much bigger impact by replacing all of your old light bulbs with fluorescent bulbs, if you haven't done that already.