Well, after reading about a dozen posts and fixes that did not work. It all boiled down to uninstalling the Samsung Driver & Kies (if you installed separately). This def works for Verizon SCH-I535 Galaxy S3 16MB with 4.0.4 load.
1. Uninstall Kies, Uninstall Samsung Driver, reboot. Reboot your phone just to be safe.
2. On your S3 select Settings-->Developer-->Enable USB Debugging (hit ok at the prompt). Don't tweak anything else here. You MUST have the Debugging turned on for phone to connect to PC.
3. Plug-in the Samsung USB cable to PC (there was some discussion on which USBs work and which don't ... so you can try different USB ports if #4 fails to work below)
4. Windows will then install whatever driver(s) it needs and whala, you can now access your S3 via Explorer...it will show up as a phone, not an actual drive.
Hope this helps and saves you the time and frustration I experienced. I never saw the little USB icon on the S3 until I wiped all the Samsung and Kies drivers from my Win 7 system & enabled debugging on phone. So the issue lies with Samsung and their crappy Kies tool...their driver does not work for S3 (i.e., the standalone download that Version posts on their site). Go with the Windows Update driver (which is auto) and you should now be able to connect via USB to your S3. Again, this works for Win 7 32bit with Android 4.0.4.
-Cheers