Go to a YouTube video – on YouTube.com

• Click “Share” then “Embed” and grab the code

Default Embed Code:

I changed the default width="560" at the start to width="100%"
Also added style="max-width: 600px;" - after allowfullscreen:

I can’t show an example unfortunately, as my Free WordPress.com blog doesn’t allow to add any code. The embed above isn’t fully responsive.