If you are paying under the table, you are in a really bad position. Have you considered the penalties involved for what you are currently doing? The consequences could change your life to the point that worrying about a nanny or day care might be small items in comparison.
Your nanny decides after some time that you abused them. They call the IRS. You are done. First, your nanny, regardless of immigration status, if that is an issue, becomes a non-issue as they will gain protections you couldn't begin to get for yourself. Then, computers will start adding up things and depending on the charges against you, your freedom could be in question.
Why do people do this? Yes, a lot of employment for nannies and such isn't reported. No, the IRS and FTB don't make it a priority, UNTIL someone files a complaint. Then, it becomes a priority because you as the employer took advantage of someone needing a job and the pile on has no limits.