The purported birth last month of the world’s first gene-edited human babies, claimed by a Chinese scientist, spurred a wave of global outrage. Scientists denounced the (as yet unconfirmed) experiment as irresponsible, and the development reinforced fears that the redesigning of DNA is moving ahead too fast and without necessary oversight.

The proliferation of similar experiments on farm animals in recent years supports those concerns. Though rapid strides have been made to map genomes—the full set of genes for humans, animals,...