What you could do, is to use for example jQuery. There are elements lower down in the structure that has unique (well unique for all List View Webparts) classes.
That code adds the class
tada-listview-webpart to the most outer element of a List View Webpart.
By running this code after the rendering of the List View Webpart, you have successfully added a class and thereby can add styling to the webpart by the class name :)
Example of running after load of
Not one hundred percent sure that
sp.js is the script you need to run after, but you can see the load order in the developer tools of a browser, and try the scripts one by one until it works.