The CSS:

.video-container {
position: relative;
width: 100%;
padding-bottom: 56.25%; /* Forces a perfect 16:9 box shape */
height: 0;
border: 3px solid #000; /* Your border goes here */
overflow: hidden;
}

.video-container iframe {
position: absolute;
top: 0;
left: 0;
width: 100%;
height: 100%;
}