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.
Ex.
$('.ms-listviewtable').closest('.ms-webpart-chrome').addClass('tada-listview-webpart')
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 sp.js
:
ExecuteOrDelayUntilScriptLoaded(function(){
$('.ms-listviewtable').closest('.ms-webpart-chrome').addClass('tada-listview-webpart');
}, 'sp.js');
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.