Yes, you're required to file a tax return when you have net self-employment income of $400 or more. As you guessed, this is because you need to pay self-employment tax.
If you're filing with TurboTax Online you'll have to use Deluxe to report your self-employment income, but you don't have to upgrade to TurboTax Self-Employed if you don't have any expenses for your babysitting income.