What do you do for ID if you don't have a car or are medically disqualified so don't have a licence?
All states issue id cards for people who don't want, qualify, or can't have a driver's license. Most ID cards look/feel the same as a driver's license and are typically issued by the department of motor vehicles or similar licensing agency, but state specifically that they are for identification purposes only.
Every state I've lived in has had photo IDs available for those without drivers licenses. They're still generally issued through the state's department of motor vehicles.
In most states, you can get a state ID that's not valid for driving, but is useful for corroborating your identity and age (and presumably residence, for those states that comply with this new Real ID stuff).
You still go to the drivers' services facility and ask them for a non-driver ID card.