The setting to Hide Alert Messages upon new mail arrival is found in the Outlook  Preferences Pane for both Outlook 2011  and Outlook 2016 . If you wish to change other application notification settings, please refer to this FAQ: How can I change my notification settings?

To Turn Off this setting:

1. Open Outlook and select Outlook > Preferences.
   
    
2. Click on Notifications & Sounds.
   
    
3.  Make sure that the For New Messages checkbox is not ticked.
   
    
4. You will no longer receive an onscreen new message notification if the box is greyed out, as below.