Hypothetical: I'm attracted to a woman who has one or more kids.
I'm going to feel more comfortable if the kid(s) is/are older, at least grade school age to mid high school. Younger than that, and the kid may get attached to you more than the mom after a while. If the kid is in late high school or later, they probably don't care whom their mom is dating, and you're just "(insert first name)".
For the, say, 6-15 year old, I would want to have some time alone with the kid, and find out what his/her interests are, which will help in picking the perfect present at a future date. I would let the kid know that I want his/her support, because if their mom has to make a choice between the two of us, she will chose the kid. You'd want the kid to feel that you are adding something positive to their life, rather than making them share their mom. Something like taking the kid to a sports event, or the museum, while mom has the afternoon off to take a bubblebath, will win you points with both. It's actually easier when it's not your own kid, because you have no disciplinary responsibilities, and you can arbitrate more impartially, if need be. You can get respect from peers based on your reputation, but you can only get respect from the young based on your actions.