Shiny is built upon Bootstrap which is able to factor in some device constraints such as using percentages as opposed to pixel values for width and height values of elements.
If you need further control over what is shown, I suggest either using a package like shinyMobile or by using a browser ready layout system such as flexboxes. Alternatively, you can use the flexdashboard package which is built so that "Components are intelligently re-sized to fill the browser and adapted for display on mobile devices." (from their website, although I have not personally used it to vouch for its ease of use)
If you want to really customize the experience and design two separate views for desktop and mobile, then you can refer to this post on how to distinguish device size.
I hope this helps, William