Yes, it's crazy that institutions allow themselves to fall so far behind. But that's the reality. Some institutions have old legacy systems that have problems with modern browsers and it's a long and difficult process to get those systems upgraded. And upgrade requests usually come standard with a lot of politics.
If you're stuck in a situation where your users are forced to use an old browser like IE6 then the reality is that it's probably best to stay with an older version of Moodle like 1.9 that doesn't struggle so much with that old browser. It's no use trying to drive the latest greatest car when there are no roads in your institution.
It sometimes helps if you approach them with a "conservative" request (i.e. don't expect them to rush the browser upgrade in a short period) and a really, really good business case for upgrading their browsers.