If I'm not using my PC, I shut it down. It cuts down on my electricity bill and gives me a warm fuzzy feeling about doing my bit to cut down on waste and help the environment.

I always leave the modem and router on as their relative power consumption is tiny compared to the behemoth that is my work machine.

You will be safe enough hooked up to the net all day if you have proper anti virus and firewall protection, though if your PC is doing nothing I don't see much point. In addition to this, the wear on your hardware won't be particularly significant if you leave it on, though I would recommend turning the monitor off, especially if it's a CRT.