Currently, using the browser interface users can only upload individual files to Connections. Users are able to select multiple files for upload, but not a whole directory or multiple subdirectories for upload.
Enable uploading of Directories and Subdirectories into Connections through the browser interface. Switching to the Windows/Mac Plugin for this is not an option.
This should be possible according to multiple sources on the web, including the W3C’s Web Platform Incubator Community Group (WICG).