If you zoom into a webpage containing an applet, you'll discover that the applet does not scale.  It remains the same size and a solid-colored rectangle will appear around it. 

Unfortunately, fewer people have their browser set to 100% zoom any more.  It's a consequence of larger monitors and higher native resolutions.  Everything will appear tiny unless you adjust the DPI settings and if you increase DPI, you'll also have to increase the zoom level in web browsers to compensate.

Is there anything we can do within the applet or within the page containing the applet to scale the applet size appropriately based on zoom and DPI settings? 

IIRC once zoomed, the getWidth() and getHeight() of the Applet container should reflect the new size, thus allowing you to use the new values to resize your game.