This is a quick user script that hooks into the 'desk.agent.case.detail' state and clicks the note button as soon as it is available.

Note.user.js

// ==UserScript==
// @name         Note
// @namespace    http://www.desk.com/
// @version      0.2
// @description  This script defaults new interactions in Desk.com Next Gen Agent to note rather than reply.
// @author       Thomas Stachl <tstachl@salesforce.com>
// @match        https://*/web/agent*
// @grant        none
// ==/UserScript==

(function() {
  'use strict';

  if (!angular || !ds) {
    return;
  }

  angular.element(document).ready(function() {
    angular.element(document.querySelector('html'))
      .scope().$on(
        '$stateChangeSuccess',
        function(event, toState, toParams, fromState, fromParams) {
          if (toState.name === 'desk.agent.case.detail') {
            var counter  = 0;
            var interval = setInterval(function() {
              counter += 1;
              var element = document.querySelector('a.note');
              if (element) { element.click(); clearInterval(interval); }
              if (counter > 5) { clearInterval(interval); }
            }, 500);
          }
        }
      );
  });
})();

In the last update I added a counter because we want to make sure intervals are cleared in any case. This update has been tested in my development environment but let me know if there is any issue with it.