I recently made a script that I think could be very useful to people, It works by you attaching the label "Docs" (which will be created by the script), to a message which you wish to copy the attachments of. Once messages under that label are found the script will copy the attachments over. I also added a feature that makes nested labels(inside "Docs") as folders inside your Google Drive - labels nested inside the nested labels will also be created inside the folder, and so on and so forth. The UI is pretty crude but it gets the Job done. Feel free to give it a try.
Please report and bugs you find.