Your income from watching the elderly gentleman is considered self-employment income and should be reported on a Schedule C. You will have to pay self-employment taxes on your net income. You are allowed to deduct any expenses that you may have incurred in your babysitting gig.
Refer to the TurboTax article How do I report income from self-employment? for instructions on how to enter your income.
For additional information, please review the TurboTax Help article Reporting Self-Employment Business Income and Deductions.