Here’s more on the episode (and hey, I know one of the writers! I’ll tell him about this thread): Children of Time (episode) | Memory Alpha | Fandom
Horatius:
…Were I in charge of such a situation, I’d make a rule that breeding in at least the first few generations must be as diverse as possible, since we can’t rely on later generations retaining the knowledge of genetics needed to avoid such problems.
So, yeah, you can marry whoever you want, but your kids will be produced via some sort of artificial breeding, so that each woman has kids with as many different men as possible, and each man impregnates as many women as possible. No kid in the first generation should have any more than half-siblings. Maximize the genetic diversity as much as possible, as long as possible.
This is discussed and similarly treated in the pretty good post-apocalyptic novel The Last Ship by William Brinkley (very different from the TV series of the same name), in which a U.S. Navy destroyer with both male and female crewmembers survives a full-on global nuclear war.
As it happens, I started a thread on this question awhile back: After Doomsday, what is the smallest possible genetically-viable human population?