Targeting the whole page
First, ensure that you’re calling body_class() inside of the HTML <body> tag:
<body <?php body_class(); ?>
Then, you can take advantage of the post format-specific classes output by body_class() to style according to your needs.
Single blog posts use the class single-post-format-{format}
body.single-format-aside {}
Archive index pages for a post format taxonomy term use the class term-post-format={format}:
body.term-post-format-aside {}
That should let you target the body, or use it as an ancestor selector to target anything else on the page.
Targeting the post container
First, ensure that you’re calling post_class() inside the post container:
<div id="post-"<?php the_ID(); ?> <?php post_class(); ?>>
Then, you can take advantage of the post format-specific classes output by post_class() to style according to your needs.
Posts with a post format assigned use the class `format-{format}’
div.format-aside {}