Ideally she would give you her social security number for you to claim it. What she does about reporting that income is up to her and doesn’t affect what you do.
If you enter the childcare credit you have to provide the Social Security number or Tax ID of the babysitter. The babysitter is responsible for reporting that income on her own tax return. The IRS matches that information up.
Babysitter will pay self-employment tax forSocial Security and Medicare on the income as well as ordinary income tax depending on how much she earned.
You have not mentioned where the babysitting occurs. If the sitter comes to your home and you have paid over $2300 in 2021 you were supposed to treat her like a household employee and withhold SS and Medicare. If she babysits in her own home or somewhere else, she is just self-employed.
https://ttlc.intuit.com/community/tax-topics/help/who-is-considered-a-household-employee/00/26770
It's not under the table anymore if you do that. You must give your care provider's name, address and social security number, and the IRS will look for matching income on their tax return. It's pretty much the total opposite of under the table. This is a discussion you should have had with the provider before you started, they would probably have charged you more if they knew they were going to be paying taxes.