The images (<img> tag) on the web has an onerror-event that fires if the image in the src is either an invalid image, or the image is not found (404). Every Domino server has an icons-folder. The most used icon/image is ecblank.gif, which Notes uses in many nasty ways on the web.
To check if a notes-server is online, you simply make an image tag with src=http://serverdomain.com/icons/ecblank.gif
onerror, change some text, pop an alertbox up etc.
In my simple demo, I add a css class to a a div, and replace() innerHTML with a status.
Demo (view source to see how it is done)
If you want to automate the testing of online-status on your company's servers, you can use the Windows API's UrlDownloadToFile ( example ) to download ecblank.gif. OnError - log error, send mail etc.