Yes, you are expected to pay taxes on income earned by babysitting. See pages 2-3 of Publication 525, attached below. To report income from childcare, you would include the money you earned on a form called Schedule C and attach it to your normal Form 1040 tax return. I've attached a copy of Schedule C below for your convenience.
Robert Hoffman is a tax attorney licensed in California. The information presented here is general in nature and is not intended as a substitute for legal advice. This posting does not create any attorney-client relationship with the author. For competent advice about your particular situation, consult your own attorney.
Mr. Hoffman has provided excellent advice. I would like to add that the Family should either provide you with a 1099 or a W-2.
This answer does not establish an attorney-client relationship, Moreover, this attorney is Licensed to practiced law ONLY in LOUISIANA and answers to questions from other jurisdictions or states are meant to provide only general information. Users should contact a local attorney in their jurisdiction or state.