Based on my comment you should clarify what you mean with sync Core Data to Dropbox.
Sync is a quite complex arguments.
If your goal is to store the entire persistent store file (e.g. sqlite store) in Dropbox Sync Api is the way. You can save the file through API. BUT be aware that the store could corrupt, so avoid it. It's a bad idea to share the store file across multiple users.
If you need to sync data across multiple devices (and not files) Dropbox does not provide this functionality. By the way Ensembles Framework by Drew McCormack provides a way to sync Core Data apps through Droppox, iCloud, etc..
Otherwise you have different options: 1) use iCloud as is, 2) set up your server with your own API to sync data (http://www.raywenderlich.com/15916/how-to-synchronize-core-data-with-a-web-service-part-1) or 3) use ready backend services like Parse or BaasBox. The former is a service where data are stored in Parse servers while the latter can be used with your own server...