22% of households in Manhattan own a car. There are about half a million households below 60th St. So there are about 100,000 cars in lower Manhattan that belong to residents. Of those, about 25,000 are used to get to work each day. The rest sit in garages.
So no, residents will not mostly be the ones paying the costs.
But suppose they were. So what? Sounds fair to me. We don't make the subway free for residents. Why should it be free to drive and store your vehicle just because you're a resident?