Call of Cthulhu by Chaosium has a notorious reputation for ending characters' careers frequently. What advice do you have for getting players to care about their characters, despite knowing they likely won't survive with their sanity intact?

Give them a back story that they have some say in. Give them some "normal life" time before you steep them in the weird. And when you do start the weird do it slowly at first and then hit them with something meaningful, a valued NPC is the target, or attack the characters character in a fashion that draws the player in and make them have to find a way of clearing their name and good character.