How do I add classes to items in Owl Carousel 2?

You can do it this way:

$(function(){
    $('.loop').owlCarousel({
        center: true,
        items:5,
        loop:true,
        margin:10
    });

    $('.loop').on('translate.owl.carousel', function(e){
        idx = e.item.index;
        $('.owl-item.big').removeClass('big');
        $('.owl-item.medium').removeClass('medium');
        $('.owl-item').eq(idx).addClass('big');
        $('.owl-item').eq(idx-1).addClass('medium');
        $('.owl-item').eq(idx+1).addClass('medium');
    });
}); 

Listening to the event of your choice

List of available events here

In the demo, I just add the class big to the main item, and the class medium to the previous and next one. From that you can play with whichever css property you want.

LIVE DEMO

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NOTE:

You could also just play with plugin classes, .active and .center (or you can also define your own, as you can see here: custom classes