It has been 51 years since the first birth-control pill became available in the U.S., bringing dozens of contraceptive options for women in its wake. There are currently two options on the market for men—vasectomy or a condom.

The research pipeline is full of possibilities, though. Scientists at Columbia University Medical Center have halted sperm production and started it again in mice with no apparent side effects, using a drug that blocks receptors for vitamin A. An injectable synthetic substance, in Phase 3 clinical trials...