Per https://developer.mozilla.org/en-US/docs/Web/API/WindowBase64/Base64_encoding_and_decoding and http://en.wikipedia.org/wiki/Basic_access_authentication , here is how to do Basic auth with a header instead of putting the username and password in the URL. Note that this still doesn't hide the username or password from anyone with access to the network or this JS code (e.g. a user executing it in a browser):
$.ajax({
type: 'POST',
url: http://theappurl.com/api/v1/method/,
data: {},
crossDomain: true,
beforeSend: function(xhr) {
xhr.setRequestHeader('Authorization', 'Basic ' + btoa(unescape(encodeURIComponent(YOUR_USERNAME + ':' + YOUR_PASSWORD))))
}
});