Add a permission setting to control who can view the "Staff online" widget.
Staff online is not a separate widget - it is part of the "members_online" widget.
You can turn off the display of Staff online globally via the widget settings, but that means nobody can see it - not even administrators.
You can turn off the display of the entire Members online widget (including Staff online) for specific users using the widget display condition field - but that shows or hides both parts of the widget.
On our sites, we want to discourage users from contacting staff directly for support and want them to use our helpdesk and report management system instead.
Using this permission setting, we can disable the display of Staff online for registered users while keeping it visible to administrators and moderators so the staff know which other staff are online. This also allows us to display the Members online part of the widget to all users, independently to who can view the Staff online component.