Hi Jane,
I've tried all the steps posted, including the re-install with the most recent installer and deleting all previous versions. When I try to delete the Dropbox app from the finder menu, I still get the message that Dropbox is open. I also had selected that I prefer Dropbox in the Finder Menu as opposed to the app, which may be at the root of the error, but there is no way to change it now that the app will not open.
Just FYI, I have a Early 2015 Macbook and just installed Mac OS ****** Sur and had just transitioned to backing up my Mac on Dropbox but now the Dropbox app itself will not open and there is no backup of my Mac in my Dropbox account online. I have also deleted and re-deleted Dropbox from my Login Items menu, but a version keeps appearing there, unchecked. I have submitted a ticket with Dropbox Support, but if you can guide me through the steps of another solution so the "open" message will not continue appearing so I can get the app to delete and hopefully re-install, I would greatly appreciate any assistance. Thanks so much for your time!