Use basic authentication with jQuery and Ajax

Use the beforeSend callback to add a HTTP header with the authentication information like so:

var username = $("input#username").val();
var password = $("input#password").val();  

function make_base_auth(user, password) {
  var tok = user + ':' + password;
  var hash = btoa(tok);
  return "Basic " + hash;
}
$.ajax
  ({
    type: "GET",
    url: "index1.php",
    dataType: 'json',
    async: false,
    data: '{}',
    beforeSend: function (xhr){ 
        xhr.setRequestHeader('Authorization', make_base_auth(username, password)); 
    },
    success: function (){
        alert('Thanks for your comment!'); 
    }
});

Or, simply use the headers property introduced in 1.5:

headers: {"Authorization": "Basic xxxx"}

Reference: jQuery Ajax API

How things change in a year. In addition to the header attribute in place of xhr.setRequestHeader, current jQuery (1.7.2+) includes a username and password attribute with the $.ajax call.

$.ajax
({
  type: "GET",
  url: "index1.php",
  dataType: 'json',
  username: username,
  password: password,
  data: '{ "comment" }',
  success: function (){
    alert('Thanks for your comment!'); 
  }
});

EDIT from comments and other answers: To be clear - in order to preemptively send authentication without a 401 Unauthorized response, instead of setRequestHeader (pre -1.7) use 'headers':

$.ajax
({
  type: "GET",
  url: "index1.php",
  dataType: 'json',
  headers: {
    "Authorization": "Basic " + btoa(USERNAME + ":" + PASSWORD)
  },
  data: '{ "comment" }',
  success: function (){
    alert('Thanks for your comment!'); 
  }
});

Use jQuery's beforeSend callback to add an HTTP header with the authentication information:

beforeSend: function (xhr) {
    xhr.setRequestHeader ("Authorization", "Basic " + btoa(username + ":" + password));
},