The Dropbox API does actually have an "App Folder" access level which does just that - it grants access to a single folder within dropbox just for that app, instead of full access to your whole dropbox. It's had the feature for a while, and I'm amazed that more apps don't use it.
I can understand a file manager or content sync app needing full access, but it's just what something like Moon+ should use!
I have E-mailed the creator of Moon+ and requested that he add support for "App Folder" dropbox access, instead of "Full". Here's hoping he is willing to implement it!