We use Fitvids in combo with Video filter to make youtube videos responsive. You can either add the JS yourself or use the Drupal module version, it works either way.
In the case of using the Drupal module, you'll add the video filter HTML selector to the UI for fitvids and then your video will be responsive.
For example, on my site, I have a video using video filter:
[VIDEO::https://www.youtube.com/watch?v=y0bA_KH-D-I]
... and then in the video filter UI, I have added the slector for the video here:
/admin/config/media/fitvids
In my case video filter is using embedded-video so I added .embedded-video to the fitvids UI in Drupal.
For more info see: