The IRS requires their addresses, Social Security or employer identification numbers, and all payments they receive from you to be reported in order to claim the dependent care credit.
If the nanny refuses to give you the identifying information, you should report on Form 2441 whatever information you have (such as the name and address). Enter “See Attached Statement” in the columns calling for the information you do not have. Then attach a statement explaining that you requested the information from the care provider, but the provider did not give you the information. Be sure to write your name and social security number on this statement. The statement will show that you used due diligence in trying to furnish the necessary information.
https://www.irs.gov/publications/p503/ar02.html
[Edited: 1-23-18|2:20PM]
If the babysitter refuses to give you her SSN, IRS allows you to claim the credit, but you must mail your tax return and state that the babysitter is refusing and that you made attempt to collect the SSN from her.
According to the IRS Publication 503, , Provider Refusal: “If the provider refuses to give you the identifying information, you should report whatever information you have (such as the name and address) on the form you use to claim the credit. Enter "See Attached Statement" in the columns calling for the information you do not have. Please attach a statement explaining that you have requested the information from the care provider, but the provider refused to furnish that information. Be sure to write your name and social security number on this statement. The statement will show that you used due diligence in trying to furnish the necessary information. "