The Drag and Drop functionality relies on the HTML5 File API and Drag-And-Drop modules,
which means that it works in modern browsers only: Firefox 4+, Google Chrome, IE10, Safari 5+. So one option is you can check whether the File Api is available in your browser and if it is not available you can hide the drop zone and display a custom message. Please check the following code I tried which works fine at my end.
$(".async-drop-zone").html("<strong>Your browser does not support Drag and Drop. Please take a look at the info box for additional information.</strong>");