Thanks for the reply. None of the "it might be this" reasons in that post seem to be relevant, e.g. I am not syncing with mobile or photos, and I am not sharing or redirecting any folders. Also, nothing has changed since dropbox not asking for this and now its asking for it.
Drobox seems to be running fine even though I hit Deny, which makes it even more suspect.
Re: Mac: "drobox would like to control this computer using accessibility features"