Dropbox if you already have an account?

Your account is allredy authenticated. For new Authentication use Reautehticate.

How can the account be authenticated if I haven’t entered my dropbox login details into the plugin anywhere?

No, when you click the Authenticate button you will send to the Dropbox page. If you Loggtin there Dropbox will use the allerdy loggtin account and only displays a page to allaw the BackWPup App. If you ther click ok Drobox will send you back to Your Blog. That is the Authentication workflow that must be done for Dropbox (oAuth). Account creditals will not be savet in BackWPup. BackWPup will get a special key from Dropbox so that it can have acsses.

No, I do not have an Authenticate button. All I have is:

-Not authenticated!
-Create Account (link)
-Reauthenticate (sandbox)
-Reauthenticate (full drop box)

When I try to click a reauthenticate button I get an error message
Dropbox API: (59) Unknown cipher in list: followed by a bunch of letters

ah, now. You can use the Beta Version from https://github.com/inpsyde/backwpup or install a mini Plugin from https://marketpress.com/wp-content/uploads/2014/03/backwpup-disable-ssl-verify.zip to fix this.