I don't have much to say on this topic, but a few people have popped up saying things about how none of the characters lead stable enough lives to raise children, including Hawke, and I'm fairly certain that's incorrect.
Consider the span of the game, about seven years, and consider the number of quests that you actually go on. In reality, adventuring is more of a weekend hobby for Hawke and co than anything else. Hawke is really just a noble who occasional kills stuff, Anders is really just a healer who every once in a while takes a hike up Mt. Sundermount to see (and kill) a mature dragon, et cetera. Also consider that Hawke and co are better prepared for emergencies than anyone else in Kirkwall. Their kid gets trampled by a horse? Papa Anders, cast regroup (or Hawke could use that grenade that revives people).
An especially amusing case, imho, is Fenris, who spends most of his time in a huge mansion with nothing to do but wait for Denarius to attack. Now, he may not be emotionally prepared for fatherhood (that child would have to get used to walking around the shards of broken wine bottles), but he has far more than enough time to handle raising a child, especially if it was born in-between acts (so it was three years old by the time you went back to gameplay).
But all of this is off-topic, because the OP wasn't talking about you having a child, he was talking about something like this: in act 1 you come across a ten-year-old beggar child, and you can give him some money, and some advice. Next act, if you told him he can only trust himself, he's a pickpocket in the docks; if you told him that there were organizations that might be willing to take care of him, he shows up in the chantry; if you told him that he needs to stick together with his friends, you see him leading a group of urchins in lowtown. By act III he's become old enough to be useful to you. Thief!Kid can be asked to steal some plot item, Chantry!Kid is a Templar recruit and has some small part to play in one of those quests, and Leader!Kid and his gang show up to back you up in some quest or other.