If Uber's costs will go up if they are no longer able to classify their employees as contractors, they can simply raise their prices. Uber appears to provide both a lower price and higher quality taxi service than traditional cabs, so even if their prices go up they will still be competitive.
If they have to raise prices above traditional cabs (highly unlikely), that suggests their service depended too heavily on a legal loophole that traditional cabs did not have.
I don't think you realize how much it would damage their business model's profitability to have to pay payroll taxes on all their drivers. It's not just about offering a better service to consumers; Uber/Lyft would emphatically not have this many backers if their business model weren't insanely profitable. If profit margins shrink, poof, there goes funding.
Anecdotally, Lyft/Uber are so much better that they could charge me a lot more and still be preferable, though obviously they expect the margins would be smaller. I doubt it would be business-model-killing smaller though.