Now you can easily clean your BuddyPress site’s activity stream anytime you need. Now new WordPress update provides Activity menu in your Dashboard’s left side bar. Visit Admin’s Dashboard, click Activity. It displays all your BuddyPress activity stream posts. You can select all or some posts and delete them permanently, just like you do in All Posts and All Pages list.
In-case if you are using older version of BuddyPress then you can follow the second method. Where, first you Visit your cPanel, Open your BuddyPress site’s database via PhpMYAdmin, Find out wp_bp_activity table and click Empty button. That’s all