How to add wrapping elements & classes to a table in a view?
You can add css class to the whole view (go to your view edit page -> Advanced -> Other -> Css Class). It will print something like this:

&lt;div class="view view-test view-id-test view-display-id-page table-responsive view-dom-id-47b9f610df77a3f6181e64ca36b9881e"&gt;
  &lt;div class="view-content"&gt;
    &lt;table class="views-table cols-2"&gt;
       ...
    &lt;/table&gt;
  &lt;/div&gt;
&lt;/div&gt;


In views UI there is no option to add class to &lt;table&gt; element, but you can create/copy a template file (e.g. views-view-table.tpl.php, you can copy it from views module, look in "Theme information" in your view for more specific details) in your theme, clear caches and edit it as you wish.