This area tends to fall under the employer's duty of care. If something happens then and there are failings then the employer may be accused of failing this duty of care.
Also from a personnal experience just asking someone if they have a driving licence is not sufficient. One place I worked at an employee claimed he had a driving licence but when I inspected it it was an International Driving Licence issued in Ghana, legal for the first year of the persons stay in this country. He had been working for the company for three years!
Where it is done, the usual arrangement is for a company to inspect the driving licence once a year, however I have heard of companies, where their employees do a lot of mileage, inspecting every quarter.
Finally think of the person whose job relies on him driving, what are the tempations if he loses his licence, does he admit it and get fired, or keep quiet and hope he doesn't get caught?