Teaching financial responsibility to your kids is very important. There are two really distinct groups; your younger children and your older children, your teenage children.
Young children, you are not going to teach financial responsibility to a two year old, so don't even try. But when they are 6 or 7 or 8 and you are doing back to school shopping, absolutely.
A couple of ways to teach financial responsibility is, one, to live it. If the family is not being financially responsible, the children don't have a chance at being financially responsible, so it's not "Do what I say, not as I do." You have to be financially responsible in your life and teaching financial responsibility to a young child -- let's say 8, and you are doing back to school shopping -- is rather than taking them back to school shopping, taking them from store to store, and listening to them complain about how they want more expensive jeans and shoes. Give them a budget on the front end. Say, "We are going back to school shopping next week. It is going to be so exciting. You know what, you have a budget of whatever you think is reasonable, say $400. Tell the child, "You have $400 to spend."
You will see a marked difference in how you have done back to school shopping. They will now run from to store to store, and turn over tags. They will stop whining at you and start talking to each other, looking for bargains, and wanting to spread out that money. That's how you start teaching financial responsibility to younger children.