what person/group in the government is in charge of the welfare system?

If you want to suggest a new law, then write to the House Ways and Means Committee - google it. If you want to complain about welfare abuse in general, I doubt that you have anything to say that they haven't heard already. Otherwise if you know of a specific abuse case contact the agency making the payments.

Agencies who make payments to beneficiaries who commit fraud would love to have the staff needed to investigate and uncover frauds. Maybe your letter could suggest that staffing to welfare agencies and Social Security be increased since it would mean great savings in avoiding overpayments. That wouldn't do any good either but the more letters they get from the public suggesting an increase in staffing just might help.