Check if current user has post in post type and is author role

Instead of using $chelseaRole ==1 it should be replaced with something like this !current_user_can('manage_categories') since the author role cannot manage categories as seen here: https://codex.wordpress.org/Roles_and_Capabilities

Polished it would look like this:

function chelsea_user_has_posts($user_id) {
  $result = new WP_Query(array(
    'author'=>$user_id,
    'post_type'=>'ait-dir-item',
    'post_status'=>'publish',
    'posts_per_page'=>1,
  ));
  return (count($result->posts)!=0);
}

function cross_check_user_items() {
  $user = wp_get_current_user();

    if (chelsea_user_has_posts($user->ID) && !current_user_can('manage_categories')) {
      echo '<style type="text/css">
               #adminmenuwrap ul#adminmenu li#menu-posts-ait-dir-item.wp-has-submenu ul.wp-submenu li:nth-child(3), body.post-type-ait-dir-item div.wrap h2 a.add-new-h2, #woocommerce-product-data.postbox div.inside div.panel-wrap ul.product_data_tabs li:nth-child(8) {display:none !important;}
             </style>';
  } else {
    //show them the goods
  }
}
add_action('admin_head', 'cross_check_user_items');