How to add a css class to this php code

You can do something like this:-

<div class="YOUR-CLASS">
<?php posts_nav_link(' — ', __('« go back'), __('keep looking »')); ?>
</div>

See Documents Here.

Hope this will help you..

There's a jQuery way for issues like this, when you're not able to add classes to WP functions. Just add the class via jQuery:

HTML

<div class="navigation">
  <?php posts_nav_link(' — ', __('« go back'), __('keep looking »')); ?>
</div>

jQuery

$(function() {
  $('.navigation a:first-child').addClass('your-class');
});

This will add a class to the first link in the div.

There's a CSS way as well without adding a class but properties:

.navigation a:first-child {
  /* your properties here */
}

If you want to add css classes to the <a> tag generated by posts_nav_link() function, another way of doing it is using these hooks in your functions.php file

add_filter('next_posts_link_attributes', 'posts_nav_attributes');
add_filter('previous_posts_link_attributes', 'posts_nav_attributes');

function posts_nav_attributes() {
    return 'class="btn btn-primary text-uppercase"';
}

Read more of these here: https://developer.wordpress.org/reference/hooks/next_posts_link_attributes https://developer.wordpress.org/reference/hooks/previous_posts_link_attributes