If a family member I live with is admittedly responsible for the disappearance of my personal property, would they be liable to pay for it?

You don't say how the family member was "responsible" for your loss.  Did they steal the property?  Were they negligent in looking after it?  Did they destroy it out of spite?  Was it jointly owned property that they used or transferred without your permission?  The answers to these and other questions can impact whether your family member is indeed legally responsible for your loss.  However, if someone is legally responsible for your loss, they are liable for it, regardless of whether they are related to you or not.  I have, unfortunately, litigated many cases involving family disputes.  Before you sue this family member, you need to think about whether there is another way to resolve your dispute, or whether it is worth taking the risk of ruining your relationship in order to be reimbursed for your monetary loss.