Is it permissible to take out a loan from a bank to help family members in their home country who are facing difficult living conditions, to enable them to start a small business, given that their private income is not sufficient to cover their basic needs?

Borrowing with interest is one of the gravest forbidden acts, and it is not permissible to borrow with it to assist parents and siblings except in dire necessity. If you are able to provide the necessities for your parents and whatever is available for your siblings, then you have fulfilled your obligation. It is necessary to urge siblings to strive for lawful earnings, to rely on Allah, and to seek forgiveness, for whoever fears Allah, He will provide for him and make a way out for him.