How to implement a Dropbox-like functionality for authenticated users to upload files?
Check out the filedepot module.


  The filedepot module is full featured Document Management module that
  has a google docs like feel. It fulfills the need for an integrated
  file management module supporting role and user based security.
  Documents can be saved outside the Drupal public directory to protect
  documents for safe access and distribution.




It works great as a ftp-like solution which allows users to upload files, except that it has a UI and is accessible the same way as default drupal pages. Lots of configuration available so you can limit the actions of users. You can also integrate organic groups with filedepot and create groups and folders available only to certain groups, if you want to get into private files for users.

or create a webform for authenticated users only and give them a file upload widget. Then you can browse the submissions in a tabular format. This allows users to upload files but not browse them if you set the permissions up properly. They will still be able to see their upload of course.