1. Manage your money in a responsible way.

Create a simple budget that works for you. Practice good budgeting habits - know how much you have to spend before you spend it.
By spending within your self-made budgeting guidelines, you will demonstrate a confidence and security about money that your kids will notice and hopefully emulate later in life.
Make a habit of saving. If you don't follow a budget, don't save money and you frequently overspend, you will have a great deal of stress about money which your kids will easily pick up on.
Which lesson do you want to teach your kids about money? Money is stressful? Or, money is to be respected and commanded in a responsible manner, bringing a sense of confidence and calm?

2. Directly teach them about budgeting and saving money early

Some experts disagree about the effectiveness of allowances and money for chores, etc. - but don't get distracted by this.
How your kids 'get' money is quite a different thing from what they do with it once they have it.
Help your kids to know the pros and cons of the different things they can do with their money.
Kids aren't generally known for having a long attention span, especially when the topic is boring, so keep your message simple.
Encourage them to save a portion of any money they get; a great idea for reinforcing this is to match each dollar they save.
Also, gradually help them to understand the different places money flows to in the adult world.
For example, explain the sales tax on the receipt for the shirt they just bought - that it is used to pay for roads, schools, etc.
Or, if you want something a little more fun for them, playing a board-game like Pay Day or the computer game The Sims can help a lot by giving them a frame of reference for understanding some of the basics of where money goes.
However, actively helping your kids to build a habit of saving money whenever they receive it, in the real world, is probably one of the best things you can do to help them build good money skills.
Explaining the pitfalls of using credit is valuable too (as well as minimizing your own use of it!), but emphasizing savings is probably more effective at a young age since it's something they can do now.
Teach them that by having a lifelong habit of saving money, they won't need to borrow as much, allowing compound interest to work for them and not against them.
When your kids get their first part-time job, continue to extend your reinforcement for saving, but now add in a proper budget.
After showing them the basics of listing their expected income along with their desired expenses and savings goals, reward them for creating their first budget.
Be creative in offering various incentives for them to stick to their budget.
By building on the savings habit you taught them earlier on, and introducing them to the concept and value of budgeting, you will have done a great deal to help your children grow into one day being responsible adults.