I know this comes a bit late but I going to leave here in case some one needs a different alternative from the above answer.
We add the function below to our theme most probably under functions.php or any other included file. The function below wraps the oembed inside the video-container div and below function is the respective CSS for the new class
/**
* Add Response code to video embeds in WordPress
*
*/
function ns_embed_html( $html ) {
return '<div class="video-container">' . $html . '</div>';
}
add_filter( 'embed_oembed_html', 'ns_embed_html', 10, 3 );
add_filter( 'video_embed_html', 'ns_embed_html' ); // Jetpack
// CSS
.video-container {
position: relative;
padding-bottom: 56.25%;
height: 0;
overflow: hidden;
max-width: 1200px;
margin: 0 auto;
}
.video-container iframe, .video-container object, .video-container embed, .video-container video {
position: absolute;
top: 0;
left: 0;
right: 0;
width: 100%;
height: 100%;
}