Add wrapper to only youtube videos via embed_oembed_html filter function

$url will contain the full URL to the embed source. E.g.: https://www.youtube.com/watch?v=_UmOY6ek_Y4, so you have to search within $url to see if the provider’s name appears:

add_filter( 'embed_oembed_html', 'wpse_embed_oembed_html', 99, 4 );
function wpse_embed_oembed_html( $cache, $url, $attr, $post_ID ) {
    $classes = array();

    // Add these classes to all embeds.
    $classes_all = array(
        'responsive-container',
    );

    // Check for different providers and add appropriate classes.

    if ( false !== strpos( $url, 'vimeo.com' ) ) {
        $classes[] = 'vimeo';
    }

    if ( false !== strpos( $url, 'youtube.com' ) ) {
        $classes[] = 'youtube';
    }

    $classes = array_merge( $classes, $classes_all );

    return '<div class="' . esc_attr( implode( $classes, ' ' ) ) . '">' . $cache . '</div>';
}