I too had this problem after I upgraded to 4.3. I'm a noob to the android world on a Verizon S3 and don't have a custom ROM running, so keep that in mind when using this. I found the following worked perfectly for me. Who would have thought you would need to install a Samsung driver on your PC...
This is how I enabled USB under 4.1.2
Unplug phone from PC.
Enable Developer Options:
MENU>SETTINGS>ABOUT PHONE>
Scroll down to the "greyed out" "BUILD NUMBER"
and tap it 5 times (possibly 7), you will see little
alerts telling you how many more times to tap.
Now go back to:
MENU>SETTINGS>DEVELOPER OPTIONS should be visible now!
Under developer options settings, check the box "USB DEBUGGING"
Next download the Samsung Galaxy USB drivers onto
your PC.
Download Center: Download Software, Firmware, Drivers & Manuals | Samsung
Choose your model and carrier.
Install and reboot.
Now when you connect your Galaxy it will either auto open
Windows Explorer (if PC), or if not, pull down the taskbar
on your droid, and under USB connection, you can select
Media Transer option.. MTD I think? or MTP. I can't
verify because it doesn't appear anymore after the 4.3 update
Give it about 30 seconds to connect, and it should appear
in Windows My Computer list of available drives.. SCH-1535 I believe. if Verizon
https://community.verizonwireless.com/message/1047241