Guilt and plausible disaster scenarios are a good start, but nothing teaches like a real disaster barely averted through heroic efforts. It took us a couple of years to convince the powers that be that our tape libraries needed replacing. It took far too much effort and we didn't get what we needed (we had to settle for SDLT320, couldn't afford LTO), but at least it was better. Each time TPTB get a wild hare about must be fully replicated! Must have a hot-backup site! we duly budget out what it would cost, and every time they decide they can't afford it.
Except the most recent round. For a wonder they decided the middle-tier storage array could get live replication. The top tier is... going to wait for its replacement, if at all. The bottom tier fell off the back of the budget.
It takes constant effort, and is very wearying over the long haul. So far we haven't had an educational disaster yet.
More generally it takes education and perseverance. Education as to risk factors and likelihood of same (also known as 'risk management', they're supposed to teach that kind of thing in business schools), as well as mitigations and costs. If you have to go there, drawing up a detailed report of exactly how screwed the company is if a likely disaster case arrives and how much money would be lost by not addressing the problem is a good start. That's doing their job for them, but sometimes you have to in order to get the right thing done.
Unfortunately, it still won't help if they decide you're just scare mongering and ignore you. At least you now have a paper trail showing you knew about the problem, tried to mitigated it, and was denied, in the event of an actual disaster. Have a near miss? Trot out that report again and see if the rose color has rubbed off their glasses yet.