Should he lend money to his sister when he knows that she is going to pay back the loan with haraam money?

Praise be to Allah.

It is not permissible for your husband’s sister to get the unemployment benefit mentioned, because she has gone against the conditions of entitlement. The money that she has taken is haraam, and she has to return it to its rightful owners. But there is nothing wrong with your husband lending money to her, even if he knows that she is going to pay back the loan with this money, because he is taking the money in a manner that is acceptable according to sharee‘ah, and the sin is on the one who acquires it; the loan is the responsibility of the borrower and there is no deal concerning the haraam wealth itself.

Dr. ‘Abdullah ibn Muhammad al-‘Imraani was asked about a man who lent someone some money, and he wants to ask for it back now, but the person who borrowed it is going to pay it back with haraam wealth. He replied: The borrower has to pay back what he owes of debts with whatever he earns in a permissible manner. The lender has the right to ask the borrower to repay the debt without asking about how he acquired the money. 

If the borrower gives the lender money that is haraam in the way in which he acquired it, it is permissible for the lender because the way he is acquiring it is different, and the sin is on the borrower. 

This may be understood from the hadeeth of Bareerah (may Allah be pleased with her), when she received some meat as charity, and she gave it as a gift to the Prophet (blessings and peace of Allah be upon him), who did not accept charity, but he accepted it because it came to him as a gift. See the report narrated by al-Bukhaari (1495) and Muslim (1074). 

And Allah is the source of help.  

End quote from Fataawa Mawqi‘ al-Islam al-Yawm (www.islamtoday.com).

Your husband should advise her and remind her. If he can refuse to take her haraam money so that she will stop, that would be good. 

And Allah knows best.