How to trigger JQuery event when drop down select list is changed in CiviCRM?
You cannot always predict the id or name for custom field for an activity, it is either custom_385_14(incase of saved custom field for activity where 14 is id from custom value table) or custom_385_-1 incase of new one.

I always follow below snippet to work incase of custom field and it does call my change event when i change option from select dropdown. I believe you choosed wrong id to actually call change event.

CRM.$(function($) {
  $("div#customData select[data-api-field='custom_385']").change(function() {
    $("#subject").val($(this).val());
  });
});


Or 


  Try after installing the extension
  https://github.com/pradpnayak/org.civicrm.jsoncaseactivity . I have
  added the JS in
  https://github.com/pradpnayak/org.civicrm.jsoncaseactivity/blob/master/templates/CRM/Case/Form/Activity.extra.tpl


HTH

Pradeep