When you began your arrangement with the babysitter did you have a clear understanding that you would be using the childcare credit on your tax return and would need their SSN or Tax ID?
If the babysitter just does not want to give you a SSN, they can easily get a Tax ID instead.
https://www.irs.gov/individuals/international-taxpayers/taxpayer-identification-numbers-tin
https://www.irs.gov/businesses/small-businesses-self-employed/how-to-apply-for-an-ein
But if the problem is that they do not want to report the income on a tax return you can follow these steps----and likely start looking for a new babysitter.
When a childcare provider will not give you a SSN or Tax ID
If they still refuse to provide an EIN or SSN, send your provider a W-10 (Request for Taxpayer Information and Identification Number). You can get a copy of the W-10 by clicking here: W-10 Form & Instructions Complete as much of the form as you can and send it via certified mail (after making copies for yourself).
If the provider does not respond within 10 business days, you can still file your return. However, you won't be able to e-file. Print and mail in your return, along with an explanation and a copy of the W-10 you sent your provider.