Get wordpress taxonomy slug name(s) to use as div class

@Ambitious Amoeba’s answer works. But have you looked into the post_class function? It’d do what you want, and save you a lot of trouble. Just use this as your div opener:

<div <?php post_class('box-item cocktails-box'); ?>>

where you pass the classes you want applied to all posts as a list or array, and let WordPress handle the rest (it’ll add classes for all taxonomies that the post is in, and you can filter it to add additional class depending on the view, if you want).