In some countries it is mandatory that the au pair has a driving license.
Anyway, it is always an advantage if the au pair possesses a driving license. Many tasks within the family -but not all tasks- require driving a car. Of course, the driving license is of little or of no use if the au pair does not have the required driving experience.
You as a host family
If the au pair is required to drive, you need to consider the following:
Clarify if the au pair has driving experience. Many au pairs do have a driving license, but they have never driven a car after the driving test.
The driving license or an International Driving Permit might be valid in the host country only for a certain time period. Please enquire about the provisions concerning this subject at your local police station and ask if maybe the au pair has to take an additional test.
Clarify with your insurance who will bear the damages which might be caused by the au pair.
You might want to pay for some driving lessons so that your au pair gets acquainted with the highway code in your country. If the au pair comes from a country where a car is driven on the other side of the road than in your country, it is advisable that the au pair takes driving lessons before he/she drives your car.
You as an au pair
As an au pair you should enquire early enough if your driving license is also valid in your host country or if you can get an International Driving Permit. Please ask the competent authorities.
Au pairs in America - USA
If you would like to work as an au pair in America (namely in the USA), you should in any case have a driving license and have enough driving experience. When you apply to one of the authorized au pair agencies in the USA, you will be asked if you have an entry in the central index of traffic offenders. So drive carefully and do not risk complaints about your way of driving.