First, if you look at the comment classes, you will notice a class called
byuser. That class is one of the default classes added by
comment_class. That is all you need. You can style that with CSS however you want.
If you theme is not using
comment_class as it should be then...
Look in your theme's
comments.php for a function named
wp_list_comments. In Twenty Twelve it looks like this:
wp_list_comments( array( 'callback' => 'twentytwelve_comment', 'style' => 'ol' ) );.
callback part is the important part here. If there is a callback in your theme's function call, that is what you need to alter. If there is no callback the WordPress default will be used so there should be no problem.
comment_class should already be in use.
The theme should have a function named the same as that callback's value in the code above. For Twenty Twelve it is
twentytwelve_comment which is defined in
functions.php. That formats and displays the actual comment list. All you should need to do is duplicate/edit that function and add the
comment_class function, right around here.