A household that employs a domestic worker for more than 24 hours a month is an employer in law, with the same registration duties as a business. There is no exemption for private households and no minimum wage below which the duty falls away.
Two registrations apply. UIF, through uFiling as a domestic employer, within 14 days, using form UI-8D. And COIDA, with the Compensation Fund, within seven days, which covers your worker if they are injured while working for you. That second one is the one households almost always miss, and it is the one that protects you personally: an unregistered employer can be held liable for the costs of a workplace injury that the Fund would otherwise have carried.
PAYE is usually a different matter. You only have to register with SARS if your worker earns enough to be liable for normal tax, which domestic wages rarely reach. If yours does, the 21 business day clock applies to you the same as to any other employer.
You still owe a payslip every pay period regardless of which registrations apply, showing gross pay, each deduction separately, and the net amount.