Hi Mark
You would need to customize the app a bit using Javascript to add in a basemap control. Here's an example Shortlist in which a basemap control has been added: Rockville Historic Buildings Catalog
We don't build basemap toggle/pickers into the Story Map app templates because the aim of these apps is to reduce the amount of UI presented to end-users. We want the story map authors as much as possible to make all the cartographic decisions to provide a good looking, simple end-user experience. But that's not to say there are specific situations (or customers/users you are building apps for) demanding more end-user control. So the ability to customize the apps is there so devs can fashion the UI they want.
One intriguing approach can be seen used here: Explore Kentucky's Tourism Regions
Kentucky made a hybrid basemap that changes appearance as the user zooms in, and in the highest zoom levels they pass seamlessly from a topographic map down to high-res imagery, no extra UI needed. Keep zooming in and the beautiful contours and hillshading of their state topo map suddenly disappears before your eyes to reveal that state's green fields and forests in full color!
