Solomon's answer about bootstrap inspired me to add the CSS the bootstrap solution uses, which works really well for me.
.iframe-embed {
position: absolute;
top: 0;
left: 0;
bottom: 0;
height: 100%;
width: 100%;
border: 0;
}
.iframe-embed-wrapper {
position: relative;
display: block;
height: 0;
padding: 0;
overflow: hidden;
}
.iframe-embed-responsive-16by9 {
padding-bottom: 56.25%;
}
<div class="iframe-embed-wrapper iframe-embed-responsive-16by9">
<iframe class="iframe-embed" src="vid.mp4"></iframe>
</div>