Its a significant problem with the usefulness of carriers today. The short answer is to keep carriers a long way from shore so that launching flood attacks of missiles can only be done from platforms such as bombers, ships and submarines. And then to focus on keeping the launch platforms away. This would be necessary as simulatenously shooting down 200 missiles is beyond the capability of even an entire carrier task force.

And the other answer is to take out the eyes and communications mechanisims of any opponent so they are unable to target a carrier with flood attacks. The missiles need to know where to go. The aim being instead to take out reconnaisance aircraft or anything else that can target it. A lot of faith is placed in this by outsiders and perhaps the USN itself. But realistically that's going to be hard to ensure over the longer term. Sensor technology and its platforms is progressing rapidly driven by the improvements in electronics. In the deep open ocean a carrier can hide. But closer to shore there are so many detection methods, and they are so cheap that its unlikely a carrier could stay hidden against an enemy that had spent a lot of money developing detection resources. Something as simple as flooding parts of the ocean with drones has no counter measure. And it should not be forgotten that many of the missiles themselves do not need precise guidance. They're perfectly capable of seeking out and targeting the carriers on their own, or even co-operatively once they're in the correct area.

The other strategy is to have aircraft on the carriers capable of very long range strike missions, so carriers are able to hit land targets while themselves being out of range of shore base missile batteries. This has been an ongoing problem for the Navy as their older long range dedicate strike aircraft became obsolete and were not replaced with similar capability. The Navy now relies on Hornets, and eventually F35C's to do this role. But their range is not sufficient to keep the launching carrier outside the range of shore based missile batteries. This means carriers cannot be used for ground attacks against even medium sized opponents without unacceptable risk.

A longer term solution is the X-47 derived unmanned strike drone. This will be highly capable against ground targets and will have much longer range allowing carriers to stay outside of shore based missile batteries.