If the person is providing child care services in their own home (they are an independent care provider) then you are not responsible for their tax situation. Pay in cash or check, it doesn't matter, but get receipts either way. You will need their tax number (social security number or federal EIN) to claim child care credits on your tax return. You don't issue a 1099-MISC to them for the money you pay them.
If the person is providing care in your home exclusively to your family (which the word nanny makes me think) then they are probably your employee. You are required to pay them the agreed wage. You don't have to withhold income tax but you do have to collect and pay household employment tax, which is the same function as social security and medicare tax in a regular job. Household employee tax goes on your tax return with your other taxes and credits. You also have to give them a W-2 at the end of the year, and you need their SSN in order to file that W-2 with the social security administration.
See here for more https://www.irs.gov/publications/p926/ar02.html
