Version 1.2 of Pythonista includes the official Dropbox Python SDK as a built-in module. Because of the way OAuth works, using it can be a little daunting at first, so I've created a minimal example that uses the webbrowser module for logging in and the keychain module for storing the credentials.
The script then just shows some information about the logged-in account, but once you're there, downloading and uploading files is relatively easy.
If you make sure that the script's name contains no spaces, you can also import it from other scripts and use the get_client function from there, so that you don't have to duplicate all that login boilerplate code everywhere you want to use the dropbox module.
https://gist.github.com/4034526
Please note that you need to create a Dropbox app in order to get this to work. Go to
http://dropbox.com/developers/apps
to create one.