The development of technology that allows human genes to be edited has stirred tremendous excitement about the potential for treating debilitating and life-threatening diseases. The technology could lead to drugs that would treat cancers and other diseases that currently are incurable.

But another facet of this breakthrough has many scientists and others worried: the possibility that the genetic makeup of sperm and eggs could be edited so that diseases that can be inherited won’t be passed on to children yet to be born. One...