I got my 66-year-old sister a mountain bike, and she's quite excited about it. She says, however, that she has some arthritis in her hands, and it hurts to brake when going on a long downhill.

My guess is that this will get better as her hands get stronger.

Anyone else have experience with this?

I wonder whether adjusting the brakes so that they are less tight would help -- that is, her hands would be less open when applying the pressure.

Those are all good questions for her MD or a PT if you know one.

Personally, in early season or after a sedentary spell I get aches and pains in the joints that have not transmitted much force lately. As my muscles and tendons tone back up, I feel better. It's possible that if she rides some more she'll improve. But I'm only 56, and I wonder if this recovery I experience would happen as quickly as I get older?

Ibuprophen or other NSAIDS can help me, too.

I'll second Road Fan's advice; having said that, I can speak to this a little from personal experience.
I'm only 58, but do have pretty advanced osteoarthritis (genetic predisposition). It first noticeably showed up in my thumbs/hands a few years ago.
I'm otherwise very fit for my age (largely as a result of cycling), but I do have trouble with any kind of caliper brake (road, canti, or v-brake), even using extemely good components, careful setup etc. I simply can't manage the grip effort required for any sustained braking, or for that matter trigger-shifting (repetitive motion problems) etc.
Brake quality, setup etc. can all help, of course, and may even resolve the problem completely for your sister, but in my case the eventual solution was/is (for me) discs -- problem solved, due to the very low effort required to exert braking force.

I'm a big fan of disk brakes too. Good grip, even in the wet, and they don't have to be smashed to get really good grip.

I've also switched to a barrel grip shifter, so I'm not doing the trigger thing, and I've got grips that have wings coming off them to provide palm support while riding.

It may be a bit of a pricey solution if your sister is a casual rider, but hydraulic disc brakes take such a light effort it's hard to believe.

One other consideration, and it's to do with bike experience and bike confidence

Just possibly, someone new-ish to riding (your post suggests that's the case, sorry if it's not) might use the brakes more than someone who is happy with gravity?

In which case, a bit of finger muscle development from riding more, and a bit of 'letting go' from riding more might ease the problem a little.

Yes, I think that's the case. She said she had the brake on the whole time on her driveway, and it's not that steep.