There are many ways to detect, using javascript, whether a browser is mobile or not.
It's easier to detect a mobile browser - and thus indirectly desktop browsers - because if a browser is not mobile, it's desktop.
Check
this link out. There are heaps of free code samples on the web.
Once you have a javascript function to detect whether a browser is mobile or not, you can then run some javascript on DOM load to hide the div with the background image.
Forexample, using jQuery:
$(document).ready(function () {
if (isMobileBrowser()) {
$('bk_image').css('display', 'none');
}
});
Hope this helps :-)