DropItToMe is a free service that works with Drop Box to allow people to upload files to your Drop Box account without giving them access to the contents of your Drop Box. 

After connecting the two services DropItToMe provides a that you can give to your students.  This will enable them to upload files into your DropBox account.  You chooase a password for this file and only give it to the students you want to be able to upload files.

I use DropItToMe so that my students don't clutter my email box with the papers they are turning in.

http://www.dropitto.me