Do drivers have to enforce seat belt use to the passengers?
The Guru Take
Not by law
There are so many areas that need to be discussed, in order to properly address this question.
Legally? Not so much
Uber and Lyft do not seem to have policies around drivers enforcing seatbelts on the passengers. With that said, drivers and passengers all have to abide by local traffic laws. It's the responsibility of each person in a vehicle to follow the rules. While these types of traffic laws often vary by state and local areas, generally, individuals are not responsible for enforcing laws on other people. This means that it's not the legal responsibility of a driver to enforce the law in their vehicle.
"In taxis and liveries, drivers are not responsible for passengers that do not comply with the law, although, as drivers, they now have to wear seat belts themselves." (source)
Watch out for Liability
With that said, drivers do have responsibilities for the safety of their customers. Hence, if there is an accident, there may be a liability on the drivers who did not make an effort to have riders buckle in.
"Drivers are always responsible for guests traveling with them or for anyone for whom a ride is requested. In the event of an accident, it is possible a passenger can hold one of the drivers liable. This isn't a certainty, but, doesn't mean that if you're injured while riding in an Uber or Lyft vehicle, you have no legal recourse." (source)
...but "enforce"? ...or deny service?
More on Driver Responsibility
This reminds me of the similar RideGuru response that was a lot to do with liability and driver responsibility. It's a great read.
Lyft encourages drivers to "make sure" but stops there...
This is what Lyft says on their driver training material:
Make sure everyone wears a seatbelt
Drivers should only take as many as passengers as there are seat belts in the car. Bigger groups of passengers should split into two cars, or use Lyft Plus, which can seat up to 6 passengers.