You’re talking about the toolbar, not the dashboard. (“Dashboard” refers to the main page of the admin.)
You can hide the toolbar on the front-end of the site for lower-powered users like so:
// show admin bar only for admins and editors
if (!current_user_can('edit_posts'))
{ add_filter('show_admin_bar', '__return_false'); }
You’ll also want to redirect users’ logins so that after they’ve successfully logged in they’re returned to the current page (or their front-end profiles, or whatever else you want) instead of being sent to the admin.