How can I add a new CSS class to a widget?

Go to your function.php and find your widget code

You must add or edit that class parameters in your sidebar code like that 'class' => 'new-class',

It would be like that:

register_sidebar(array(
  'name' => __( 'Right Hand Sidebar' ),
  'id' => 'right-sidebar',
  'class'       => 'new-class',
  'description' => __( 'Widgets in this area will be shown on the right-hand side.' ),
  'before_title' => '<h1>',
  'after_title' => '</h1>'
));

You must check out Wordpress Codex for more usage

Don't edit your parent themes functions file.

register_sidebar(array(
'name'          => __( 'Sidebar name', 'theme_text_domain' ),
'id'            => 'unique-sidebar-id',
'description'   => '',
'class'         => 'add class here',
'before_widget' => '<li id="%1$s" class="widget %2$s">',
'after_widget'  => '</li>',
'before_title'  => '<h2 class="widgettitle">',
'after_title'   => '</h2>' );
));

Copy it over to your child themes functions and make the changes there or simply register a new widget.

You could then hook it in from your child themes functions like this

add_action( 'your_hook', 'new_widget' );
function new_widget() {
if ( is_your_conditional_tag() && is_active_sidebar( 'new-widget' ) ) { 
dynamic_sidebar('new-widget', array(
'before' => '<div class="new-widget">',
'after' => '</div>',
) );
    }
}

You can also add more classes to the dynamic_sidebar function.