How do we get an 88-year-old man with beginning dementia to wear adult diapers?

I have never called them "diapers" around him. I refer to them as underwear. Thanks for the advice.

First of all, don’t call them “diapers”, especially to his face. I’ve been told by more than a few medical professionals that this is insulting to the person who needs to wear them. Diapers are associated with babies.

There will still be much clean-up even if he does wear incontinence briefs. But, if he is still fairly aware of the issue, make sure he’s there when you do the clean-up. Do so without comment, but make sure he sees you dump and rinse his underwear and clean up. Mention to him that his wearing Depends would make things much easier. You can also replace his cloth briefs with the incontinence briefs.

Have you asked him to wear the Depends and he refused? If this is the case and you're having to clean him up after each accident, convey to him that since you're the one doing the clean-up, refusing the Depends isn't an option. He has to pitch in and do his share of the work = wearing Depends.