Drupal's Service module provides resources to implement these kind of APIs. You can use Service module and enable its in-build login/logout resource.

Please follow these steps to implement login validation using JS from Phonegap, Sencha or any other remote application:

1: Install and enable Service module.

2: Create a service endpoint and add its path.

3: In resource tab select Checkbox for User (login and logout).

4: Enable REST Server Response Formatters & Request Parsing

5: For testing a Resource from Drupal Services, install poster in firefox and check your added service:- Tool->poster

URL: http://localhost/my_drupal_site/?q=/system/connect.json

Content Type: application/x-www-form-urlencoded

Actions: POST

6: If connection establish then process to step 7 otherwise check you connection.

7: Create the simple php login page in remote application and call following javascript to make an ajax call:

$('#page_dashboard').live('pageshow',function(){
  try {
    $.ajax({
      url: "http://10.0.2.2/my_drupal_site/?q=my_services/system/connect.json",
      type: 'post',
      dataType: 'json',
      error: function (XMLHttpRequest, textStatus, errorThrown) {
        alert('page_dashboard - failed to system connect');
        console.log(JSON.stringify(XMLHttpRequest));
        console.log(JSON.stringify(textStatus));
        console.log(JSON.stringify(errorThrown));
      },
      success: function (data) {
        var drupal_user = data.user;
        if (drupal_user.uid == 0) { // user is not logged in, show the login button, hide the logout button
          $('#button_login').show();
          $('#button_logout').hide();
        }
        else { // user is logged in, hide the login button, show the logout button
          $('#button_login').hide();
          $('#button_logout').show();
        }
      }
    });
  }
  catch (error) { alert("page_dashboard - " + error); }
});

This is just a jQuery code and can be used in any remote application which use jQuery. I hope this will help you.