How can I add a CSS class to the body tag based on the current path alias?

In your theme, you can add the following code in template_preprocess_html:

$current_path = Drupal::service('path.current')->getPath();
$path_alias = Drupal::service('path_alias.manager')->getAliasByPath($current_path);
$path_alias = ltrim($path_alias, '/');
$variables['attributes']['class'][] = DrupalComponentUtilityHtml::cleanCssIdentifier($path_alias);