You may think, "What's so different? I'm dating her, the babysitter is watching them". Nothing could be further from the truth. No matter what the children's ages you'll have to relate to them sooner or later, and whether you do that successfully or not may determine whether you get to continue dating that woman with children or not.
Even if your date is a woman who doesn't have full custody of her children, or whose children are old enough that they only come home on college vacations and summer breaks, that woman and her children come as a package. You date the mother, you must relate to the children. That's a fact.
Of course, it's important that your date's children like you. That may prove more difficult than you expect, depending on their ages, and the length of time since the father walked out - or was thrown out - of the picture. Looking like a potential competitor to the father that that child daydreams will come back home and make a family again, is not going to make you popular.
On the other hand, especially if it's a male child, the longer the father is gone, the more they begin to think of themselves as "the man of the house" and the more they resent having to share the attentions of their mother with you.
The best thing you can keep in mind as you think of how to date a woman with children is not to try too hard to make a good impression with the kids. Don't be phony, don't talk down to them, and don't buy them lavish gifts and toys. Just be yourself, put yourself out there gradually as a potential friend and after many successful dates with the mother it's okay to suggest an outing with the kids, and occasional a small gift. Perhaps a fast game of basketball or scrabble, or reading to a small child while mom finishes getting ready for a night on the town can break through the barriers the child or children have built faster than any phone "Let's be best friends" gesture you can make.
Children have more smarts about your relationship with their mom and their mom's relationship with their dad than you might think. When the opportunity arises comfortably tell them you're happy they get to see their dad often. Tell them you're open to a good friendly relationship with them, that you love their mother (if you do) and that you'll never try to replace their father in their hearts.
