The best way I can think to do that is to have the professor create a folder and then invite the students as "uploaders" which is the most limited of the Box access types. Here is the description:
Uploader: An Uploader is the most limited access that a user can have in a folder and provides limited write access. A user assigned uploader will see the items in a folder but will not be able to download or view the items. The only action available will be to upload content into the folder. If an Uploader uploads an item with the same name as an existing item in the folder, the file will be updated and the existing version will be moved into the version history.
Here is a list of all available access types.
Hope that helps but please let me know if you still have questions or if there is anything else I can do to help!