California Gov. Jerry Brown has until the end of September to sign legislation that would require public companies based in the state to put a certain number of female directors on their boards. Before picking up the pen, he should give weight to how ineffective these quotas can be.

Such quotas have been snaking their way through Europe’s boardrooms for more than a decade. Researchers say they can help boost morale among aspiring female business leaders and potentially boost profits.

But...