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:

    <div class="view view-test view-id-test view-display-id-page table-responsive view-dom-id-47b9f610df77a3f6181e64ca36b9881e">
      <div class="view-content">
        <table class="views-table cols-2">
           ...
        </table>
      </div>
    </div>

In views UI there is no option to add class to `<table>` 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 and edit it as you wish.