Adding class to featured image

You can add custom class to post thumbnails. the_post_thumbnail accepts array of attribute $attr where you can specify image class.

So your code will be.

<div class="home-featured-img">
    <?php
        if ( has_post_thumbnail() ) { // check if the post has a Post Thumbnail assigned to it.
            the_post_thumbnail( 'full', array( 'class'  => 'responsive-class' ) ); // show featured image
        } 
    ?>
</div>

In above code, responsive-class is the custom class I have added, you can change it as you require.