Make a custom plugin for your site, put this code in it.

add_filter('oembed_dataparse','oembed_youtube_add_wrapper',10,3);
function oembed_youtube_add_wrapper($return, $data, $url) {
    if ($data->provider_name == 'YouTube') {
        return "<div class='whatever'>{$return}</div>";
    } else {
        return $return;
    }
}

Note: The filter runs when the data is gathered from the oembed provider. So, it you already embedded the post, and the embed data is cached, this won't modify it.

Make a custom plugin for your site, put this code in it.

add_filter('oembed_dataparse','oembed_youtube_add_wrapper',10,3);
function oembed_youtube_add_wrapper($return, $data, $url) {
    if ($data->provider_name == 'YouTube') {
        return "<div class='whatever'>{$return}</div>";
    } else {
        return $return;
    }
}

Note: The filter runs when the data is gathered from the oembed provider. So, if you already embedded the post, and the embed data is cached, this won't modify it.

Make a custom plugin for your site, put this code in it.

add_filter('oembed_dataparse','oembed_youtube_add_wrapper',10,3);
function oembed_youtube_add_wrapper($return, $data, $url) {
    if ($data->provider_name == 'YouTube') {
        return "<div class='whatever'>{$return}</div>";
    } else {
        return $return;
    }
}

Make a custom plugin for your site, put this code in it.

add_filter('oembed_dataparse','oembed_youtube_add_wrapper',10,3);
function oembed_youtube_add_wrapper($return, $data, $url) {
    if ($data->provider_name == 'YouTube') {
        return "<div class='whatever'>{$return}</div>";
    } else {
        return $return;
    }
}

Note: The filter runs when the data is gathered from the oembed provider. So, it you already embedded the post, and the embed data is cached, this won't modify it.