Does that allow you to connect using IE but the Ajax interface or does it return some errors? In the image the server shows as QLICK. Is that in the DNS server and resolve all right from the computers that show the errors? If you are running IIS instead of QlikView Web Server, do the user have the right permissions?
I am not 100% sure but you can try to reinstall the IE Plugin. I mean you can use other browsers (Chrome or Firefox) to uninstall and then reinstall the IE Plugin from IE. I had same problem few days back and I got it working with this method.