I want to load in a new class but only if the current page is single-movies.php

You should probably have a look at is_page_template() which actually can check whether a specific template is used.

So your code should be like:

<div class="sponsors <?php if( is_page_template('single-movies.php') ) { echo " movies"; } ?>">

Be aware, that is_page_template() does not work within The Loop, as written in the Codex. You may have a look at the alternative which is mentioned there.

Edit:
Added in a ; after ” movies”