How can I get the current class of a div with jQuery?

Just get the class attribute:

var div1Class = $('#div1').attr('class');

Example

<div id="div1" class="accordion accordion_active">

To check the above div for classes contained in it

var a = ("#div1").attr('class'); 
console.log(a);

console output

accordion accordion_active

Simply by

var divClass = $("#div1").attr("class")

You can do some other stuff to manipulate element's class

$("#div1").addClass("foo"); // add class 'foo' to div1
$("#div1").removeClass("foo"); // remove class 'foo' from div1
$("#div1").toggleClass("foo"); // toggle class 'foo'