if you could make it to something like this:
<div class="videoWrapper">
<iframe width="600" height="338" src="https://www.youtube.com/embed/f5CcOq8UzkI?feature=oembed" frameborder="0" allowfullscreen=""></iframe>
</div>
then responsive css for that would be like this:
.videoWrapper {
position: relative;
padding-bottom: 56.25%; /* 16:9 */
padding-top: 25px;
height: 0;
}
.videoWrapper iframe {
position: absolute;
top: 0;
left: 0;
width: 100%;
height: 100%;
}
if you have no control on the markup, then what we can do is when you put that shortcode in the editor, add some div around it like this:
<div class="videoWrapper">
</div>