Lyft and Uber both perform background checks on drivers. Lyft also has a mentorship program where new drivers are screened for safe driving, car quality/condition, and other criteria by an experienced driver.
What's insufficient about the background checks Uber already performs? If they're quantifiably less safe than other transportation mechanisms, customers will vote with their wallets.
I'd be opposed to adding additional regulation if not needed.
Uber also fought against thorough background check legislation in texas if I remember correctly.
There are tons of other regulatory issues like insurance.