You can create new folders within a project directly from Finder. They are created as Docs & Files folders in Basecamp.
Create a folder
- Right-click inside the project folder in Finder and choose New Folder.
- Give it a name right away.
- Drag files inside to upload them to that folder.
Note: Name the folder when you create it, rather than leaving it as "untitled folder" and renaming it later; this way it appears with the correct name in Basecamp from the start.