Hi,
The min-width is not going to control the height of the row at all. The element containing the menu itself and that has the background attributes must have it's height set to auto in order for it to grow with the content. The default height of all elements is auto but more than likely the menu container element has a specific height set. It's very hard to determine what element(s) need their styling adjusted without seeing the skin. While the demo site has many artisteer skins that behave consistently with your description, they don't all have the same markup and style so that doesn't help much. Can you give us a link to your site so we can try to figure something out for you?
Thanks,
Joe D.