In your layout, add this before any other JS runs:
<script type="text/javascript"><!--
  function authToken(){
    return '<%= form_authenticity_token if protect_against_forgery? -%>';
  }
//--></script>
authToken is coded as a function so that it's less likely you'll accidentally overwrite it with other JavaScript.
In your main JS, you can then call authToken(), and it'll return your authenticity token as a string to include in your Ajax calls. For example:
$.ajax({
  type: 'PUT',
  url:  url,
  data: {
    foo: bar,
    authenticity_token: authToken()
  },
  complete: function(data){}
});
Note that if you use Rails' built-in form_for helper, it automatically adds the authenticity token in a hidden input. If you want to send all of the form's data, including the hidden auth token, you can simply use:
var $form = $('form');
$.ajax({
  url:      $form.attr('action'),
  type:     $form.attr('method'),
              // "get" or "post"; overridden by Rails' hidden "_method"
              // input value, e.g., "put"
  data:     $form.serialize(),
              // Includes hidden "authenticity_token" and "_method" inputs
  complete: function(data){}
});
This pattern is often useful when you've already written a form that works without JS, and you're adding an unobtrusive layer of JS that simply sends the form's data via Ajax.