To make the youtube video iframe responsive, just force a wrapping container to the desired aspect ratio and tell the iframe to fill the container.

css

.video-wrapper {
  position: relative;
  width: 100%;
  height: 0;
  padding-bottom: 75%; /* default youtube aspect ratio */
}

.video-wrapper iframe,
.video-wrapper object,
.video-wrapper embed {
  position: absolute;
  top: 0;
  left: 0;
  width: 100%;
  height: 100%;
}

scss

With some help from the css aspect ratio snippet this is very easy in sass.

.video-wrapper {
  @include box-aspect-ratio(420/315); // Just use the pixel dimensions of the "original" size

  iframe,
  object,
  embed {
    @include fill-parent();
  }
}