wp_dropdown_categories – how to save in widget?

You didn’t add the dropdown/select name attribute, that’s why it’s not working as expected. Please try updating your $dropdown_args, here’s the updated arguments –

$dropdown_args = array(
    'taxonomy'         => 'cs_categories',
    'id'               => $this->get_field_id( 'cs-category' ),
    'name'             => $this->get_field_name( 'cs-category' ), // Added this line
    'show_option_none' => __( 'Select category', 'text-domain' ),
    'hide_empty'       => true,
    'hierarchical'     => true,
    'depth'            => 2,
    'echo'             => 0,
    'selected'         => $instance[ 'cs-category' ],
    'class'            => 'widefat'
);