So many people have turned this into a 'are helmets safe or not'.
That's really arguing the wrong point - of course helmets improve survivability of a bicycle accident.
The point here is, for bike-sharing programs, should helmet use become mandatory?
The answer should be : no.
Ultimately few people are going to climb on a bike and think it is safe riding without a helmet. They will know that they are doing a dangerous thing.
The idea with bike-share programs is to get people using shared bikes to move around a city instead of either public transport or private cars. If carrying your own helmet (or using a publicly shared-helmet) is a requirement, fewer people are going to use the bike-sharing.
This is a simple argument about the increased use of bike sharing programs against the increased number of people with head injuries from that bike sharing program. I don't know what those numbers are, but it would mean higher head injuries but also higher bicycle use. The number of head injuries can only be measured by the set of people who chose to ride the bike without a helmet, who had an accident and hit their head. Some people (regular users, for example) would choose to still use helmets when riding bike sharing, so it's not like removing the law will mean nobody wears a helmet anymore. At the margin there will be more injuries, but also more bike-miles ridden. It's up to people to make a decision which is more desirable.
It might sound like a tough decision, but really, this type of decision is everywhere. Speed limits are set with the balance of people who will be injured or killed in more serious accidents, balanced with the ability for more people to get to their destination more quickly.