Set CSS class for the comment block
I would do it by adding a new template for the block. See Drupal 7 Template (Theme Hook) Suggestions

block--[region|[module|--delta]].tpl.php base template: block.tpl.php
Theme hook suggestions are made based on these factors, listed from
the most specific template to the least. Drupal will use the most
specific template it finds:
block--module--delta.tpl.php
block--module.tpl.php
block--region.tpl.php