Can an embedded video be made responsive? I currently use the Video filter module to embed Youtube videos in my posts, but on mobile devices the videos are not responsive. They still maintain their width no matter how narrow the screen is.

I am open to another method of embedding videos even raw HTML and Javascript if the video displays will be responsive
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 as its wrapper HTML element so I added .embedded-video to the fitvids UI in Drupal. 

For more info see:


https://github.com/davatron5000/FitVids.js/ 
http://fitvidsjs.com
https://www.drupal.org/project/fitvids