If the babysitter is NOT your child or your dependent, then the rule is that you are not required to file or pay household employer tax if the employee is under age 18, provided that providing household services is not their primary occupation (such as, they are a student and babysit on the side). If household services is their primary occupation, you must withhold and pay HHE taxes.
Even assuming the babysitter is a student, then from this person's birthday to the end of the year, you would be subject to the rules for withholding and paying HHE taxes. But because you don't count wages paid while they were under 18, you would only need to actually pay HHE tax on your tax return and issue a W-2 if the amount you paid after they turned 18 and before 12/31/23 was more than $2700. See table 1 of publication 926.
https://www.irs.gov/publications/p926#en_US_2024_publink100086740
(However, you will need to pay HHE tax, and issue a W-2, if they still babysit for you in 2024, if you pay more than $2700 over the year.)
Assuming the amount paid during that time was less than $2700, you don't issue a W-2 or pay HHE tax on your tax return. But, to claim the credit, you must put their name and SSN on your tax return on form 2441. The IRS will be looking for matching income on the babysitter's return.
The babysitter will report the wages on line 1b of form 1040, "household employee wages not on a W-2." In most cases, a child who earns a little money does not jeopardize their status as their parents' dependent. Whether or not the babysitter must file a tax return depends on their total financial situation, any other income they have, and whether any of that income is considered "unearned" (which could be scholarships, prizes, gambling income, hobby income, or interest and dividends from investments). It's not really any of your business.
