All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
Your wife`s sister is a non-Mahram* and her staying at your and her sister`s house entails several restrictions from an Islamic perspective. Therefore, you must lower your gaze, not mingle nor stay in private with her. In addition, she can`t show her adornment in front of you nor can be complaisant in her speech. If these are difficult to abide by, then her staying at your house is prohibited and your mingling with her in a prohibited manner is also prohibited. This since the Prophet (PBUH) told us that such things are traps set by the devil. The evidence on this is that Allah Says (What means): "O you who believe, do not follow in the steps of Satan. For whoever follows in the steps of Satan, assuredly he enjoins indecency and what is reprehensible. And were it not for God's bounty to you and His mercy not one of you would ever have grown pure. But God purifies whom He will, and God is Hearer, Knower." {An-Nur, 21}.
Moreover, Omar Bin Al-Khattab (May Allah be pleased with him) said: "And no one of you should be alone with a woman for the Shaitan will be the third one present." {Musnad Ahmad, 1/18}. Ibn Hajar commented on this Hadith: "Its Isnad is Sahih" {Hedayat Al-Rewayah}.
As for her working in a usurious bank, we (Iftaa` Department) remind her that Riba/usury is a major-grave sin, which is prohibited by scholarly consensus. Also, Allah has declared war against eaters of usury and warned that their wealth will be effaced and that they will be among the losers in the life of this world and the next. Allah Says {What means}: "Those who devour usury shall not rise again, except as one whom Satan has made prostrate from touch; that is because they say, 'Trade is like usury': God has permitted trade, and forbidden usury. Whoever receives an admonition from His Lord and desists, he shall have his past gains, and his affair is committed to God; but whoever reverts - those are the inhabitants of the Fire, abiding therein. God effaces usury, but He augments voluntary almsgivings with interest. God loves not any guilty ingrate." {Al-Baqara, 275-276}. He The Almighty also Says {What means}: "O you who believe, fear God, and give up the usury that is outstanding, if you are believers. But if you do not, then be warned of war from God, and His Messenger. Yet if you repent, you shall have your principal sums, not being unjust, and no injustice being done to you." {Al-Baqara, 278-279}.
In addition, the Messenger of Allah (PBUH) has cursed whoever helps in committing this major-grave sin. Jabir (RA) narrated: "Allah's Messenger (PBUH) cursed the one who accepts usury, the one who gives it, the one who records it and the two witnesses to it, saying: "They are all the same." [Reported by Muslim]. An-Nawawi commented on this Hadith by saying: "It prohibits helping in sin." {Shareh Muslim, 11/26}.
If her work helps directly in Riba/usury, then she (Wife`s sister) is sinful and she had better look for a lawful job. Even if her work doesn`t directly help in Riba/usury, it is reprehensible, but doesn`t reach the degree of prohibition.
As for benefiting from her money and accepting her gifts, we, and Allah knows best, are of the view that it is better to be on the safe side by avoiding such doubtful matters. However, jurists have permitted benefiting from one whose lawful money has mixed with unlawful money, even if the latter outweighs the former. This is attested to in the fact that the Messenger of Allah (PBUH) ate from the food of the Jews, accepted the invitation of the Jewish woman who presented a poisoned sheep to him as a gift and he engaged in commercial transactions with them. This is despite the fact that, as known to all, their money is mixed with forbidden usury. In addition, Allah the Almighty has described them in the following verse where He Says {What means}: "Listeners to calumny and consumers of unlawful gain." {Al-Mai`dah, 42}.
Banks` money is a mixture of the lawful and unlawful, but many of their services are lawful, such as transfers, currency exchange and the like.
Ibn Hajar (May Allah have mercy on him) said: "It is recommended for a person to provide for himself and his dependents from that which is lawful. However, it isn`t prohibited to engage in business with one whose money is mostly unlawful nor to eat from that money." {Tohfat Al-Mohtajj, vol.9/pp.389}.
It is stated in "Al-Mohathab" by Al-Shirazi: "It isn`t permissible to engage in business with one whose whole money is unlawful. The evidence on this is that Abu Mas'ud Al-Ansari narrated: "The Messenger of Allah (PBUH) prohibited the price of a dog, the earnings of the fornicator (from harlotry), and the news of the fortune-teller." It is reported that Al-Zuhari said about a whore who accumulated earnings from harlotry: "It isn`t permissible for her master to eat from those earnings because the Messenger of Allah has forbidden fornication and its earnings.
It is disliked to engage in business with one whose money is a mixture of lawful and unlawful or accept his gift. This is attested to in the following Hadith: An-Numan bin Bashir narrated that the Messenger of Allah (S) said: "The lawful is clear and the unlawful is clear, and between that are matters that are doubtful (not clear)." However, engaging in business with such person and accepting gifts from him is permissible because apparently the money is his. Thus, it isn`t forbidden to take from that money." {Al-Majmou`, vol.9/pp.417}.
In conclusion, it is permissible for you (Asker) and your children to benefit from the money, which your wife`s sister pays. However, we recommend that she look for a house to live in alone to prevent forbidden Khilwa*. And Allah The Almighty Knows Best.
* A non-Mahram is a person with whom marriage is generally permissible.
*Khilwa, in Shariah law, is an offense consisting of being caught alone in private with a member of the opposite sex who is not an immediate family member.