I tend to use the dropbox implementation from this github: https://github.com/zant95/elementary-dropbox. It integrates into the wingpanel with a matching white icon and appears to work flawlessly for me. I hate using that phrase, but literally my only problem has been that the icon does not change to black with the theme on wallpaper change, but it works.
If this does not solve the issue, it may be related to the browser - I have found that Dropbox now wants you to login using a browser window instead of authenticating through the application itself.
Hope this helps at least a little,
William