In the Online section of the Configure Site > Social Networking page, you can choose to use the Members Online feature, whether to use it by default, and whether to allow members to choose their privacy setting for showing whether they are online or not.  Like other features, you can choose the images to display for a member online / offline, and how to display the online/offline image/text.

If you use the Online feature, it will display on the Public Profile page of a member automatically, if you are using the Default page type format.