$('#u25210-bw').has( "#banner-biggest" ).addClass('my-class');

<div class="browser_width colelem" id="u25210-bw">
 <div>
   <!-- biggest banner -->
  <a href="/" target="_blank">
    <div id="banner-biggest" alt="на главной" style="background-image: url('/58c0847705f64.jpg');min-height: 367px;background-color: rgb(0, 0, 0);background-position: center bottom;background-repeat: no-repeat;background-size: cover;">
    </div>
  </a>
 </div>
</div>
<script>
if ($('#u25210-bw img').length>0) {
   $('#u25210-bw').addClass('my-class');
};
</script>

$('#u25210-bw:not(:has(#banner-biggest))').addClass('my-class');

$('#u25210-bw').find('#banner-biggest').closest('#u25210-bw').addClass('my-class');