Menus functionality allows you to add a hamburger menu to Genesis themes on mobile displays, that don’t already support one. You can fine tune how the hamburger menu operates.
Also this module allows you to add a search box to the primary menu. You may have to style the search box to suit your purposes as it will take on the styling of the theme. Sometimes this styling is too clumsy (read big) for a search box on a menu.
You can also easily add login or logout links to the menu if you have a membership site.