As others have noted, the problem is more likely to be too many children than not enough. I suspect a generation ship would have to put limits on how many children couples could have rather than trying to force them to have more children. I suppose this could be a problem if the initial crew was made up entirely of infertile people, hermits, and homosexuals. Or more realistically, if there were enough such people in any given generation along the way that population fell. But unless something about the nature of life on the ship makes people not want to have children or incapable of having children, there would likely be enough people who want to have ten children to make up for those who have none.
Trying to create unusual cultural norms for such a ship faces one tough problem: Presumably once the ship has been on its way for a couple of months, it has traveled beyond the reach of anyone back on Earth. If you had ships that could flit back and forth across the light years in a few hours or days, you wouldn't need a generation ship, you'd just send people in these FTL ships. So you can make up all the rules you like for how these people are supposed to behave, but once they're beyond your reach, they can freely chuck the rules and do whatever they want, and there's absolutely nothing you can do about it. Even if you picked the initial crew to be people who agree to this unusual society that you think is a good idea, the next generation is going to rebel against it, or at least question it. That's what teenagers do. The idea that you are going to raise children in a carefully controlled environment so that they completely, slavishly follow everything their parents teach them is a pipe dream. People have been trying to do that for thousands of years, and it's never worked. (Every now and then I hear people who have just had their first child talk about how they are going to raise this child to follow in their footsteps and uphold all their best ideals — whether that means being tolerant and open-minded or working tirelessly to stamp out all the inferior races or whatever — and I can only smile and say, yeah, good luck to you. That's not how it works.)
Maybe, possibly, you could indoctrinate the original crew with brainwashing techniques and drugs and high-tech mind control, and then they do the same to their children, generation after generation. But would a crew of such doped-up robots really be the kind of elite you need to successfully complete such a dangerous and complex mission? And even if it worked, it would take only one generation, anywhere in the 600 year voyage, to say, No, we're not going to do this to our children, and the whole plan falls apart. (There might be a science fiction story there, about a few who decide not to brainwash their children, and now there are two factions on the ship, the brainwashed and the not-brainwashed, who struggle for control.)
If your ship was really big enough for 20,000 people, I don't think there'd be a lot of trouble with maintaining genetic diversity and allowing people to choose mates in the traditional way. If the crew was much smaller, if it was only a few hundred, I can see there being an issue that if, say, Al and Betty are the only people on board with a certain gene, that if they marry each other you risk losing that gene, and so they must be forced to marry other people.
The smaller the crew the more you'd have to work to insure initial genetic diversity. You don't want everyone on board to be, say, white Anglo-Saxons descended from people who lived in or near London. You want to make sure you have some whites, some blacks, some Hispanics, some Asians, some tall people, some short people, some artistic people, some detail-oriented people, some big-picture people, etc. A society advanced enough to launch such a ship is probably advanced enough to be able to do genetic tests to be sure that there is sufficient genetic diversity. If the "first draft" crew, perhaps selected for specific job skills, is missing some genes, that they could say, Hey, we need some people with gene A17G4, which controls predisposition to growing bananas, or whatever.