Wordpress - make Wordpress image captions responsive

You're going to want to use:

@media (max-width: 988px){
  .wp-caption {
    /* Force the box to be 100% */
    width: 100% !important;
  }
  #content .wp-caption a img {
    /* Scale down if too big */
    max-width: 99.03225806%; /* 614/620 */
    height: auto;
  }
}

Another possibility is to change the shortcode output so that the width is no longer hard-coded. Modifying the Codex example to have no width:

add_filter('img_caption_shortcode', 'my_img_caption_shortcode_filter',10,3);

/**
 * Filter to replace the [caption] shortcode text with HTML5 compliant code
 *
 * @return text HTML content describing embedded figure
 **/
function my_img_caption_shortcode_filter($val, $attr, $content = null)
{
    extract(shortcode_atts(array(
        'id'    => '',
        'align' => '',
        'width' => '',
        'caption' => ''
    ), $attr));

    if ( 1 > (int) $width || empty($caption) )
        return $val;

    $capid = '';
    if ( $id ) {
        $id = esc_attr($id);
        $capid = 'id="figcaption_'. $id . '" ';
        $id = 'id="' . $id . '" aria-labelledby="figcaption_' . $id . '" ';
    }

    return '<figure ' . $id . 'class="wp-caption ' . esc_attr($align) . '" >'
    . do_shortcode( $content ) . '<figcaption ' . $capid 
    . 'class="wp-caption-text">' . $caption . '</figcaption></figure>';
}

http://codex.wordpress.org/Function_Reference/add_filter#Example

Here's a much simpler and cleaner solution:

function my_img_caption_shortcode_width($width, $atts, $content)
{
    return 0;
}

add_filter('img_caption_shortcode_width', 'my_img_caption_shortcode_width', 10, 3);