Please avoid any: 'you don't' answers :)
How do you tell people you're cool?
It's demonstrated, if it's there.
It depends on the vertical, and the people that company serves.
Speak in their language, understand THEIR problems and earn their trust.
I think the common language all businesses speak is revenue and competition. If you can show them competitors in their industry that are beating them because of good design, they'll immediately see the value in it.
What is "design" in this case, and why do you need to "convince" anybody? The way the question is phrased makes it sound like you're trying to force a preset approach onto a company instead of starting with their needs and going from there.
Depending on what you mean by "design", its value to a given company might very well be zero, in which case they'd be right to remain unconvinced.
It's going to be difficult.
I had this same issue with trying to get good design / good tech into a company that didn't care.
They didn't care because even with bad design / bad tech they were still the leading company in their industry (fashion trade shows).
Design is even harder because its subjective (don't care how many self-appointed design-gods say otherwise on DN). Even more so, if stakeholders aren't designers, than design, to them, could be seen as just an output, not a way of thinking, solving problems or presenting a solution to a business or user challenge.
While you've asked for "you don't" answers this is sort of the answer for some companies. The company I was at made it clear they didn't care so while I attempted to discuss, educate, lead by example or whatever else way I could fathom they just literally gave less than two fucks about good design or good technology.
If the team is receptive then you have a chance. If they aren't receptive and its killing your soul then I'd suggest leaving.
NOTE: I never usually suggest leaving but if you're on DN asking this question, I'd imagine you care about it more than the average bear. And in that case you might as well try your shot somewhere else where your interest in your career and field are taken seriously.
By explaining them the benefits of the user and their trust / useability. Remember Benefit over Cost is Value (B/C=V)
Is it really still a problem? You can simply point out at how businesses that implement good design are usually more successful than others, there's plenty of examples: Apple, Ikea, Netflix, even the recent change of mentality at Google...
recent change of mentality at Google
It depends ;)
From what position you try / want to convince them?