Assuming that the child is considered your employee (i.e. he is not self-employed), then from IRS Pub 501: for a person who is claimed as a dependent on someone else's return (which I assume is the case here):
You must file a return if any of the following apply.
Your unearned income was more than $850.
Your earned income was more than $5,150.
Your gross income was more than the larger of —
$850, or
Your earned income (up to $4,850) plus $300.
It also states:
Child's earnings. Amounts a child earns by performing services are his or her gross income. This is true even if under local law the child's parents have the right to the earnings and may actually have received them. If the child does not pay the tax due on this income, the parent is liable for the tax.
If you are his employer, be sure to withhold and pay the necessary Social Security tax for the child.
If the child is considered self-employed, then he must file if his earnings are greater than $400.
All of this applies to filing a federal tax return. Check your state to see what the requirements are for paying state income tax where you live.