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.

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, clear caches and edit it as you wish.