How can I detect CSS Variable support with JavaScript?

We can do this with CSS.supports. This is the JavaScript implementation of CSS's @supports rule which is currently available in Firefox, Chrome, Opera and Android Browser (see Can I Use...).

The CSS.supports() static methods returns a Boolean value indicating if the browser supports a given CSS feature, or not.
– Mozilla Developer Network

With this, we can simply:

CSS.supports('color', 'var(--fake-var)');

The result of this will be true if the browser supports CSS variables, and false if it doesn't.

(You might think that CSS.supports('--fake-var', 0) would work, but as noted in comments on this answer Safari seems to have a bug there making it fail.)

Pure JavaScript Example

On Firefox this code snippet will produce a green background, as our CSS.supports call above returns true. In browsers which do not support CSS variables the background will be red.

var body = document.getElementsByTagName('body')[0];

if (window.CSS && CSS.supports('color', 'var(--fake-var)')) {
  body.style.background = 'green';
} else {
  body.style.background = 'red';
}

Note that here I've also added checks to see whether window.CSS exists - this will prevent errors being thrown in browsers which do not support this JavaScript implementation and treat that as false as well. (CSS.supports was introduced at the same time CSS global was introduced, so there's no need to check for it as well.)

Creating the browserCanUseCssVariables() function

In your case, we can create the browserCanUseCssVariables() function by simply performing the same logic. This below snippet will alert either true or false depending on the support.

function browserCanUseCssVariables() {
  return window.CSS && CSS.supports('color', 'var(--fake-var)');
}

if (browserCanUseCssVariables()) {
    alert('Your browser supports CSS Variables!');
} else {
    alert('Your browser does not support CSS Variables and/or CSS.supports. :-(');
}

The Results (all tested on Windows only)

Firefox v31

Chrome v38

Internet Explorer 11

Set a CSS style with CSS variables and proof with Javascript and getComputedStyle() if it is set... getComputedStyle() is supported in many browsers: http://caniuse.com/#feat=getcomputedstyle

HTML

<div class="css-variable-test"></div>

CSS

:root {
  --main-bg-color: rgb(1, 2, 3); /* or something else */
}

.css-variable-test {
    display: none;
    background-color: var(--main-bg-color);
}

JavaScript

var computedStyle = getComputedStyle(document.getElementsByClassName('css-variable-test')[0], null);

if (computedStyle.backgroundColor == "rgb(1, 2, 3)") { // or something else
    alert('CSS variables support');
}

FIDDLE: http://jsfiddle.net/g0naedLh/6/