How do you style the dropdown on Google Places Autocomplete API?

If you use firebug (as mentioned in a comment to your question...) you see that the container with the autocomplete results is a DIV with the class "pac-container" and the suggestions are inside it as a DIV with the class "pac-item". so just style with CSS.

This is now documented by google: https://developers.google.com/maps/documentation/javascript/places-autocomplete#style_autocomplete