We have a housekeeper that comes two days per month. We pay by check. Should we withhold taxes from her check?

Take this with a grain of salt as I am not a tax attorney nor an accountant. However, in my dealings you are not required to withhold taxes as this woman is not your employee. In my limited opinion, she is a sole proprietor (responsible for her own taxes) or at best a contract employee (who would still be 1099'ed if she is even paying taxes).

Did you ever represent that you were withholding taxes? I do not see where she would get such an idea that you (and perhaps anyone else who paid her by check) was responsible for withholding her taxes.