I have a standalone crossbrowser HTML5 drag and drop example here: http://jsfiddle.net/rvRhM/1/
Have a look at the dragstart and dragend events. dm is the element being dragged.
EventUtil.addHandler(dm, 'dragstart', function(e) {
    e.dataTransfer.setData(format, 'Dragme');
    e.dataTransfer.effectAllowed = effect;
    var target = EventUtil.getCurrentTarget(e);
    target.style.backgroundColor = 'blue';
    target.style.cursor = 'move'; // You can do this or use a css class to change the cursor
    return true;
});
Be sure the reset the cursor when dragging ends:
EventUtil.addHandler(dm, 'dragend', function(e) {  
    var target = EventUtil.getCurrentTarget(e);
    target.style.backgroundColor = '';
    target.style.cursor = 'default'; // Reset cursor
    return true;
});