Hi E. -
As someone mentioned earlier, this is HER problem, not yours and, truthfully, this is a lesson we all need to learn from both sides.
When you a hire a babysitter to work in your home, you will not be taxing her pay directly, therefore she is an independent contractor. She becomes responsible for monitoring and paying her own taxes. On the other hand, because of your situation, it's not like you will be submitting your babysitter's Social Security number come tax time; rather, I would imagine that you will just gather together your documents and create an itemized list of your child care expenses. So, even if you claim this stuff on your taxes, the IRS would be hard-pressed to link the income to this woman on paper.
So, she should understand that what *you* do with your taxes does not automatically dictate what *she* needs to do with hers (although I guess we know the answer to that!) Personally, I wouldn't have mentioned it to her at all.
If I am somehow mistaken here, and you DID ask for a social securtiy number, well - that changes everything and I wouldn't work for you either. LOL
Good luck!