The following instructions are for disabling the Dropbox desktop alerts in Windows 10, Windows 8.1 and Windows 7.
Click on the Dropbox Icon on your taskbar
Click on the Gear Icon (Top Right hand side) and then click on Preferences
Whilst the General tab is displayed uncheck the checkbox which says Show Desktop Notifications and click on OK
These instructions are based on the latest Dropbox app as at 21/06/16 and may change slightly as Dropbox updates it’s application.