Yes, both employers and employees have to increase their productivity when the minimum wage is increased. It makes many jobs (like elevator operator) obsolete, those people have to skill up or take jobs that require them to do more. There could also become a point where skilling up/working harder is no longer feasible. In that case, a basic income starts to look a lot more appealing.

Minimum wage is set at the state and city levels. The federal minimum wage is probably obsolete in about 90% of the country right now, perhaps close to 100% now.