I had extensive mechanics for this in 1st edition because in character pregnancy and parenthood are almost inevitable in any long running campaign. Sooner or latter, some player is going to believe it is right for their character to enter into an erotic relationship with someone, and eventually in a realistic universe that's going to lead to pregnancy and children.
I've known of players who retired adventuring characters to allow them to be parents and spend time with their children. That's probably unusual, but with really good RPers or people who have narrative goals, you can have people see that as a satifying outcome and story arc. The player would argue that once you have kids, your job is no longer saving the world and becomes the more important job of saving your kids.
I think this is a matter that's entirely up to the player. I wouldn't try to tell a player how to be a parent. The relationship of their character to their child is their decision.
The DM's responce should be to try to grow the situation in a way that is a reasonable and realistic responce to the character's actions. The DM needs to treat the child as any other NPC and develop a personality for the child (I actually had rules for determining the personality randomly based on the parents alignments and the astrological sign), and then imagine how the child, the child's gaurdian/caretaker, and so forth respond to PC actions.
I'd try to avoid sterotypes. Not every child hates its parent for being absent. The missing parent can also be idealized, which can be just as big of a problem if not more so. The missing parent might also be a non-factor to the child, and any future intrusion would be considered unwelcome. Spouses might be understanding of the other parent's absence, or they might be filled with complete rage about it. Or somewhere in the middle. Decide on a personality for the child, and work with it. Of course, this is only likely to matter if the pacing of your campaign is such that children have time to grow up.
I'd pay careful attention to the historical problem of parenting and absent father's (often away on long voyages, serving in the army, etc.). Childhood is something of a modern romantic social construct, and ancient childhood is more marked by the looming prospect of death than anything else. Presumably, if the parent can tuck away the child amongst a loving larger extended family somewhere (time to invent the PC's relatives!) then the child is in fact living an almost normal life, even with a parent absent more often than not. If the parent on the other hand can't do this, then the question of where the parent tucks the kid away (convent, boarding school, foster parents, etc.) is going to be the essential one. It's going to be even more essential if the baby is a 'monster' of mixed race.
Loved ones can be great plot hooks. While infants and three year olds are basically fragile burdens, eventually they can become important NPC's in their own right. In a world where magic and heroes are available, child heroes are possible. There are real world 12 year old kings who were beserker warriors not only capable of killing grown men with battle axes, but having done so in a dozen battles, so its not at all improbable in a fantasy world that the PC's 12 year old kid is a dangerous 3rd or 4th level character in their own right.
