The best way to have children turn out to be responsible adults may be to be a very responsible adult, so they see what a responsible adult does.
So often I've heard parents talk about how they insist that their children do household chores "so they'll learn to be responsible". Sometimes these are even chores the parents have decided they, themselves, will no longer do because "the children need to learn". I'm not "anti-chores" when the matter is handled in a way that is reasonable; but sometimes parents require children to do the very chores they, the parents - whose responsibility those chores are - won't do them!
The parent who sits on the couch behind a newspaper every evening, rather than spending time with the children or even doing housework, is showing children that when people grow up they don't need to do what someone else wishes they would. The parent who sits and talks to another parent (or on the cell phone) at the park and gets so engrossed in conversation may not notice if her child needs something. I'm not saying parents shouldn't socialize, but we've all seen people not noticing the children because of being too busy in conversation with an adult. These same parents may try to teach their children that "there is a time and place for things".
When kids see their parents doing all kinds of things the kids wouldn't want to do (getting up to make breakfast for everyone else, paying bills, calling the plumber, fixing the roof, walking the dog in the middle of the night, etc.) they often not only start to see that this what being an adult is, but they also may admire their parents for doing all the things they do. (I think kids need to see that being adult can also mean having fun sometimes, but that's a topic for another day.)
When children have parents who act very responsibly, and when, as part of acting responsibly parents also talk to children all throughout their childhoods about why one thing or another is important, there's a good chance children grow up to be apples that didn't fall far from the tree.
There is one other factor that could play a role in which children grow up to be responsible adults; and that is when children have childhoods that are pretty carefree and secure, and parents who assure that their children's childhoods are safeguarded, the children often grow up not missing anything and more than ready to take on their own adult responsibilities when the time comes. It is at least possible that children who miss too much of what children need from childhood may grow up still wanting to hang onto to their childhood in some way and not too interested in anything they may view as a burden.