Your family members are not legally responsible for your individual debts after you pass away. Any debts you left behind must be paid out from your estate before the executor or administrator can distribute the remaining estate (if there are any) to your beneficiaries.
However, if your family member(s) hold any joint assets with you, such as a joint account, a personal loan, or a mortgage, they will be liable to pay off the debts as surviving co-owners.
Read more: What Happens To My Stuff When I Die? (Singapore Edition), All About Probate & Administration (Singapore Edition)