If the state sends you a check to pay an in-home child care employee, you may be considered their employer. See Guide to the Nanny Tax for Babysitters and Employers.
The funds are generally not taxable to you if you pay the employee directly. However, you may be responsible for handling payroll taxes (Social Security and Medicare Taxes) This means you need to report the wages on a W-2 and pay any payroll taxes owed. If the employee does not report the income and you have issued them a W-2, the responsibility to pay taxes rests with the employee.
For more information, you can also refer to the IRS guidelines on family caregivers and self-employment tax