|I think climbing-specific training will affect things as much as actual climbing. Given that guys like Pete Livesy was a pioneer of climbing-specific training in the UK in the 70s, it didn't really take off until later, so we've not really seen a large number of hard-training climbers get old yet. i.e. if they were 25 in 1982 they're still short of 60.
Just climbing, no training, obviously does not put as much intense force onto very particular body parts, so it's not such a surprise that plenty of people climb for 30 years and have no arthritis ascribed to it. In mountaineering though, there *are* plenty of older climbers who give up due to bad knees, though it's hard to say how much of this was natural and either pounding downhill under heavy packs made it worse, or made no difference.
In the 80s plenty of guys tweaked their elbows and forearms doing lots of chins off a rigid bar or Bachar ladder, and partly this was because they trained hard on these on their 'day off' from actual climbing. So slightly inappropriate training combined with lack of recovery time caused injuries, some of which may flare up in their later lives (if they're still alive).
Now people may train smarter, but they may also train harder and climb harder, creating very specific and intense forces on particular body parts (though mostly fingers and forearms/elbows). But most of those people are under 40, so we won't know for years just how much age, combined with any genetic tendency to arthritis, exacerbates such activity.
Of course arthritis, in whatever form, is not necessarily the only medical problem, and we probably all know older climbers who've been at it 40 years and get pains and tweaks and soreness and whatnot, most probably due to climbing, but not actually arthritis, or anything particularly serious.
Squashing feet into extremely tight rock shoes is a slightly different matter and there does seem to be quite a few people around with mangled feet after years of doing this. Whether that's a problem in old age is, again, yet to be seen for most.
Overall, compared to the general population, climbers are probably medically better off than most. Go sit in the waiting room of a hospital or doctor's office, or spend a week in hospital and you'll get a different appreciation of the medical health of most Australians.