How to make a transparent border using CSS?

You can also use border-style: double with background-clip: padding-box, without the use of any extra (pseudo-)elements. It's probably the most compact solution, but not as flexible as the others.

For example:

<div class="circle">Some text goes here...</div>

.circle{
    width: 100px;
    height: 100px;
    padding: 50px;
    border-radius: 200px;
    border: double 15px rgba(255,255,255,0.7);
    background: rgba(255,255,255,0.7);
    background-clip: padding-box;
}

If you look closely you can see that the edge between the border and the background is not perfect. This seems to be an issue in current browsers. But it's not that noticeable when the border is small.

Well if you want fully transparent than you can use

border: 5px solid transparent;

If you mean opaque/transparent, than you can use

border: 5px solid rgba(255, 255, 255, .5);

Here, a means alpha, which you can scale, 0-1.

Also some might suggest you to use opacity which does the same job as well, the only difference is it will result in child elements getting opaque too, yes, there are some work arounds but rgba seems better than using opacity.

For older browsers, always declare the background color using #(hex) just as a fall back, so that if old browsers doesn't recognize the rgba, they will apply the hex color to your element.

Demo

Demo 2 (With a background image for nested div)

Demo 3 (With an img tag instead of a background-image)

body {
    background: url(http://www.desktopas.com/files/2013/06/Images-1920x1200.jpg);   
}

div.wrap {
    border: 5px solid #fff; /* Fall back, not used in fiddle */
    border: 5px solid rgba(255, 255, 255, .5);
    height: 400px;
    width: 400px;
    margin: 50px;
    border-radius: 50%;
}

div.inner {
    background: #fff; /* Fall back, not used in fiddle */
    background: rgba(255, 255, 255, .5);
    height: 380px;
    width: 380px;
    border-radius: 50%; 
    margin: auto; /* Horizontal Center */
    margin-top: 10px; /* Vertical Center ... Yea I know, that's 
                         manually calculated*/
}

Note (For Demo 3): Image will be scaled according to the height and width provided so make sure it doesn't break the scaling ratio.