Add class to on sidebar widget

In functions.php file :

add_action( 'widgets_init', 'my_register_sidebars' );
function my_register_sidebars() {
    /* Register the 'primary' sidebar. */
    register_sidebar(
        array(
            'id'            => 'sidebar-2',
            'name'          => __( 'Main Sidebar', 'mystic' ),
            'description'   => __( 'A short description of the sidebar.', 'mystic' ),
            'before_widget' => '<div id="%1$s" class="widget %2$s">',
            'after_widget'  => '</div>',
            'before_title'  => '<h3 class="widget-title">',
            'after_title'   => '</h3>',
        )
    );
    /* Repeat register_sidebar() code for additional sidebars. */
}

In sidebar.php :

<?php
if ( is_active_sidebar( 'sidebar-2' ) ) { ?>
    <ul class="submenu">
        <?php dynamic_sidebar( 'sidebar-2' ); ?>
    </ul>
<?php }
?>

EDIT
If you are looking to modify the output of Recent Posts Widget,
you can use this Extend Recent Posts Class 😉
All you have to do is to modify it as you want 🙂

SYA 🙂