I have used Talon-systems.com ETA package from the FI perspective. I wasn't too impressed by it but it did do the job. This was several years ago so it may have improved by now. That's the expensive solution of course.
For the cheap and simple solution, find an online storage site that allows shared access to folders as suggested. Please do read the small print before signing up though, if its free or cheap, you may be signing away the rights to anything you upload and you don't want to do that with your student files.