Overriding Drupal core base class
No, you can't change the base class. This is a PHP language construct, it's not possible to override it.

I'm not exactly sure what you are trying to do but it is likely that what you are looking for is a plain old hook_form_alter() + third party settings on the block config entity.

The Block class D8 sandbox is a simple module that is doing exactly this and allows to configurabe custom CSS classes for every block.

Note that this is on a different level, you are altering the block config entity and it will only apply to how Drupal core/block.module uses block plugins but not e.g. page_manager. Doing it in a more generic way is simply not possible.