I just wasted 3 hours of my life trying to set this value in xml.
Normally you think that in a Block when you see $this->getChildrenWrapClass() that the value for that would come from xml or in an observer where that value is being set by $block->setChildrenWrapClass(), however for this you would be totally mistaken.
That was all the different types of xml I was trying to use to get that value set!
Well, it turns out this is actually a parameter that is passed into getHtml()
This is actually found on a template file
<?php $_menu=$this->getHtml('level-top','yourValueGoesHere') ?>
<?php if($_menu): ?>
<?php endif ?>
Basically, if you want your category navigation to have a div around the sub navigation <ul> you pass in that second argument. Lesson learned, not everything in Magento is XML sometimes it is just plain on PHP.