Try: (add to functions.php of your theme)
add_filter( 'woocommerce_breadcrumb_defaults', 'wpse_woocommerce_breadcrumb_defaults' );
function wpse_woocommerce_breadcrumb_defaults( $args ) {
$args['wrap_before'] = '<nav class="woocommerce-breadcrumb my-class">';
$args['wrap_after'] = '</nav>';
return $args;
}
Or, copy:
wp-content/plugins/woocommerce/templates/global/breadcrumb.php
to
wp-content/themes/your-theme/templates/global/breadcrumb.php
UPDATE: This is the correct path and not above: (thanks @mmm)
wp-content/themes/your-theme/woocommerce/global/breadcrumb.php
and change the echo $wrap_before;
and echo $wrap_after;
to your liking..
Reference: https://docs.woocommerce.com/document/template-structure/