Are there any ways to cure or prevent curses in Clouds of Xeen? I'm running through the Northern Sphinx and I don't want to Town Portal to Vertigo's temple every time a mummy doesn't like me poking through its sarcophagus.

AFAIK curses can only be removed in temples. Which makes fighting curse-afflicting monsters a pain in the ass. But if you are of a sufficiently high level, you can usually fight reasonably well even while cursed, only town-portalling a few times.

You can try if casting divine intervention works.
Getting cursed in WoX isn't as bad as in the previous might and magic games. You only get a reduced luck stat, since you can beat the game with only 1 luck there's no need for fixing instantly.

But curses (and other effects) get worse over time, right? And there's no "Suppress Curse" spell like there is for Poison and Disease.

I'm worried because I remember going through the Temple of Yak and one of my party members getting cursed, then dying later.

Curses don't get worse over time and you can't die because of them either IIRC.
I guess your cursed party member died when fighting the boss in the Yak temple whose magic attack causes death instead of sleep.
PS: Death can be healed easily.

Has he been unnaturally aged? Sometimes your characters can get really old through magical aging and then they die suddenly. But that might only happen when resting, not when walking around.

No, he certainly wasn't unnaturally aged.

I think I rested once inside the Temple before my character died. This was a couple of days ago so my memory is getting hazy.

Curses get worse over time, and if any stats (even luck) reaches 0, the character dies. I believe the death-by-zero-stat check may only occur at 5am though, I can't really remember. Encounter enough cursing monsters that keep increasing the curse, and your luck will eventually run out. Literally. And then you die.