Hi,
For elements in cornerstone, you can use the class field to add visibility classes
Classes:
x-hide-xl – Extra Large | 1200px & Up
x-hide-lg – Large | 980px – 1199px
x-hide-md – Medium | 768px – 979px
x-hide-sm – Small | 481px – 767px
x-hide-xs – Extra Small | 480px & Smaller
So if you want to hide your element in mobile you can add x-hide-xs class in the class field of your column element.
http://screencast.com/t/lHetWx8GDLN9
You can do the same for other elements.
Note: You need to enable advanced controls for the class field to appear.

Hope that helps.