It seems that you need to set the request header with the base64 auth data manually.
The instructions are here: http://coderseye.com/2007/how-to-do-http-basic-auth-in-ajax.html
You first need to take the following code from here: http://www.webtoolkit.info/javascript-base64.html to do the base64 encoding
/**
*
*  Base64 encode / decode
*  http://www.webtoolkit.info/
*
**/
var Base64 = {
    // private property
    _keyStr : "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/=",
    // public method for encoding
    encode : function (input) {
        var output = "";
        var chr1, chr2, chr3, enc1, enc2, enc3, enc4;
        var i = 0;
        input = Base64._utf8_encode(input);
        while (i < input.length) {
            chr1 = input.charCodeAt(i++);
            chr2 = input.charCodeAt(i++);
            chr3 = input.charCodeAt(i++);
            enc1 = chr1 >> 2;
            enc2 = ((chr1 & 3) << 4) | (chr2 >> 4);
            enc3 = ((chr2 & 15) << 2) | (chr3 >> 6);
            enc4 = chr3 & 63;
            if (isNaN(chr2)) {
                enc3 = enc4 = 64;
            } else if (isNaN(chr3)) {
                enc4 = 64;
            }
            output = output +
            this._keyStr.charAt(enc1) + this._keyStr.charAt(enc2) +
            this._keyStr.charAt(enc3) + this._keyStr.charAt(enc4);
        }
        return output;
    },
    // public method for decoding
    decode : function (input) {
        var output = "";
        var chr1, chr2, chr3;
        var enc1, enc2, enc3, enc4;
        var i = 0;
        input = input.replace(/[^A-Za-z0-9\+\/\=]/g, "");
        while (i < input.length) {
            enc1 = this._keyStr.indexOf(input.charAt(i++));
            enc2 = this._keyStr.indexOf(input.charAt(i++));
            enc3 = this._keyStr.indexOf(input.charAt(i++));
            enc4 = this._keyStr.indexOf(input.charAt(i++));
            chr1 = (enc1 << 2) | (enc2 >> 4);
            chr2 = ((enc2 & 15) << 4) | (enc3 >> 2);
            chr3 = ((enc3 & 3) << 6) | enc4;
            output = output + String.fromCharCode(chr1);
            if (enc3 != 64) {
                output = output + String.fromCharCode(chr2);
            }
            if (enc4 != 64) {
                output = output + String.fromCharCode(chr3);
            }
        }
        output = Base64._utf8_decode(output);
        return output;
    },
    // private method for UTF-8 encoding
    _utf8_encode : function (string) {
        string = string.replace(/\r\n/g,"\n");
        var utftext = "";
        for (var n = 0; n < string.length; n++) {
            var c = string.charCodeAt(n);
            if (c < 128) {
                utftext += String.fromCharCode(c);
            }
            else if((c > 127) && (c < 2048)) {
                utftext += String.fromCharCode((c >> 6) | 192);
                utftext += String.fromCharCode((c & 63) | 128);
            }
            else {
                utftext += String.fromCharCode((c >> 12) | 224);
                utftext += String.fromCharCode(((c >> 6) & 63) | 128);
                utftext += String.fromCharCode((c & 63) | 128);
            }
        }
        return utftext;
    },
    // private method for UTF-8 decoding
    _utf8_decode : function (utftext) {
        var string = "";
        var i = 0;
        var c = c1 = c2 = 0;
        while ( i < utftext.length ) {
            c = utftext.charCodeAt(i);
            if (c < 128) {
                string += String.fromCharCode(c);
                i++;
            }
            else if((c > 191) && (c < 224)) {
                c2 = utftext.charCodeAt(i+1);
                string += String.fromCharCode(((c & 31) << 6) | (c2 & 63));
                i += 2;
            }
            else {
                c2 = utftext.charCodeAt(i+1);
                c3 = utftext.charCodeAt(i+2);
                string += String.fromCharCode(((c & 15) << 12) | ((c2 & 63) << 6) | (c3 & 63));
                i += 3;
            }
        }
        return string;
    }
}
Then you'll need the code to construct the auth data, it's just a base64 of the user and password:
function make_base_auth(user, password) {
  var tok = user + ':' + password;
  var hash = Base64.encode(tok);
  return "Basic " + hash;
}
And then you just add the header with jQuery before making the request:
var service_url = "https://www.udemy.com/api-2.0/courses/"
$.ajax({
    type: "GET",
    url: service_url,
    dataType: "xml",
    data: "ParamId=" + FormId.value,
    processData: false,
    beforeSend : function(req) {
         req.setRequestHeader('Authorization', 
               make_base_auth ('myClientId', 'mySecret'));
    },
    error: function(XMLHttpRequest, textStatus, errorThrown) {   
         ajaxError(XMLHttpRequest, textStatus, errorThrown); 
    },
    success: function(xml) { DoSomething(xml); }
});