It is not mandatory that au pairs in the USA have a driving licence. However, as cars are the most popular means of transport, most host families will require their au pair to drive. That's why most sponsor agencies require that applicants have a valid driving licence.
Many host family need their au pair to drive their children to kindergarten, school, sports, music lessons, doctor appointments, friends, etc., as cars are the main way of arranging daily travel in many American locations. It is therefore important to clarify in advance issues of insurance and liability, which your au pair agency can also help you with.
Au pairs must check if their driving licence is valid or if they need an international licence. The sponsor agency arranging the au pair stay can provide more information.
More on driving licence regulations in the USA is available on the website Transportation and Public Works.
Information about international driving licences can be found at automobile organisations or road traffic authorities in your home country.