If you enter the childcare credit on your tax return then you also have to enter the Social Security number or Tax ID of the childcare provider that you paid. The IRS matches that information up to the income reported by the person with that identification. You need her SSN or Tax ID. It is her responsibility to report her income to the IRS on her own tax return.
Yes----my reply told you that your cousin has to claim the income she received for babysitting on her own return. Read it again. Your part is to put her SSN or Tax ID on your own return in order to get the childcare credit. Reporting her income on her own tax return is her own business--not yours.
If she does not want to give out her SSN to anyone she can easily get a tax ID from the IRS to use instead, and you can use that when you claim the credit.