If you don't save up enough money before the attack, your families dies.
However, there seems to be a little loophole in The Darkness attack. If you chose Elliot/Elise at the beginning of the game and you chose to keep them after the Kidnapped quest, they may still be alive even if you've barely any money simply because they were minor but still decently major characters in the plot.
There is a glitch where a family will disappear. It has happened to me. Family was living in Bowerstone Market. After the final battle (which went through the town and past my house), my nice husband and children were missing . The house still showed that I owned it and the map shows a heart on the house flag, but the house sign does not. If you play the female role, it is SOOO hard to find a decent husband in this game and I was upset to lose one who was nicer looking and made decent conversation to his wife.
If your family lives in Old Quarter, they will probably disappear after the first battle when you fight Logan's soldiers. It's a good idea to move them to a safer area before you leave for Aurora as you will have to fight that battle as soon as you land on your return.
Family also sometimes disappears if they live in the castle. If they happen to be in the audience when you make decrees in the throne room, they get deleted by the game when the session ends. In this case, a popup will say , "Following the death of your husband/wife your children have been sent to the orphanage." Jasper will then tell you that you can go and adopt them to bring them home again.