Drew Houston coded the first version by himself. From http://www.wired.com/business/2013/09/dropbox-2/ - "Houston started coding a version of what would become Dropbox’s signature folder, the portal to an app that magically holds and syncs the same files on every computer and mobile device where you install it...."

Also, if you look at his YC application, it is clear that he had written the first version. -> YC application => https://dl.dropboxusercontent.com/u/27532820/app.html