I am testing Site Designer and Don't see a way to add responsive hamburger options for mobile or to add drop-down menus to the base, non-templated page. I don't see anything in the elements and I thought it might be there?

Look in the "Components" Tab on the top menu of Site designer (second Option Top Left)
or
Open "FILE" then select 'Open Components Folder"
Or open the menu on this page "Store" then select "Web Components" to download any there that you like

More options here: https: mock-up.coffeecup.com . Choose your framework, and then find the section 'Navigations' and see if you spot something you like.