I had this same issue but found the following that works great:
The key to creating a responsive YouTube embed is with padding and a container element, which allows you to give it a fixed aspect ratio. You can also use this technique with most other iframe-based embeds, such as slideshows.
Here is what a typical YouTube embed code looks like, with fixed width and height:
<iframe width="560" height="315" src="//www.youtube.com/embed/yCOY82UdFrw"
frameborder="0" allowfullscreen></iframe>
It would be nice if we could just give it a 100% width, but it won't work as the height remains fixed. What you need to do is wrap it in a container like so (note the class names and removal of the width and height):
<div class="container">
<iframe src="//www.youtube.com/embed/yCOY82UdFrw"
frameborder="0" allowfullscreen class="video"></iframe>
</div>
And use the following CSS:
.container {
position: relative;
width: 100%;
height: 0;
padding-bottom: 56.25%;
}
.video {
position: absolute;
top: 0;
left: 0;
width: 100%;
height: 100%;
}
Here is the page I found the solution on:
https://www.h3xed.com/web-development/how-to-make-a-responsive-100-width-youtube-iframe-embed
Depending on your aspect ratio, you will probably need to adjust the padding-bottom: 56.25%;
to get the height right.