This issue wouldn't affect you unless you were hit by an UberX driver while the driver didn't have anyone in his or her car, so avoiding them won't make any difference. Once a passenger is in the UberX car, Uber has a million dollars of commercial insurance that applies. I believe this isn't an issue if you use one of the black cars, since those cars are commercial livery cars and the drivers carry their own insurance on top of whatever Uber provides.

http://blog.uber.com/ridesharinginsurance

The dispute is about what should be covered with no passenger in the UberX car. California is trying to regulate that, and Uber is spending a bunch of money in opposition. I think Uber should be required to insure their drivers at all times, but that's not a reason to avoid the service. The service has still done a tremendous service to the US markets that it serves, and at least in my opinion, is nicer and safer than a taxi.