Is it permissible to take from the money of a sister who works in a usurious bank, which she gives as a gift or buys something with, and is it permissible for her usurious money to mix with the father's household expenses?

You must advise the sister and warn her about the consequences of usury (riba), explaining that it is not permissible to work in a usurious bank except out of dire necessity. This is based on the saying of the Messenger of Allah, peace and blessings be upon him: "Allah has cursed usury, its devourer, its payer, its scribe, and its two witnesses," and he said, "they are all equal." These banks are at war with Allah and His Messenger, and anyone working for them is cooperating with them in sin and warring against Allah. One must avoid consuming from her usurious salary so as not to fall under the general category of those who devour usury.