Yes Uber drivers do go through background checks before they are allowed to become an Uber driver and also they go through routine background checks.
Uber is now doing a more tougher background check than they did before to make sure the drivers with Uber have no bad criminal history.
Uber wants to make sure the Uber drivers are safe for people and will get the people to and from their destinations safely without harm.
But even then if someone goes through an Uber background check and checks out good without any criminal history it doesn't always mean that the person is a good person to be driving for Uber.
So you still need to be very careful but a lot of people who drive for Uber are very good people who will do no harm to you or your family.
But you still need to be safe and aware.