First, supposing that good example is the fundamental principle. How am I responsible in daily life?
The Little Prince puts self-care and care of one's planet, one's domain, first. A child probably thinks naturally that most care is his mother's domain. It takes time and patience to widen out his "planet" of responsibility. Luckily we do have that time. Life Skills for Kids points out that even if our kids go to school daily, and given that they sleep for about ten hours, we still have about 7 hours a day, or over 50 hours a week, to teach our kids what they need to know in life.
To get a better idea of what life responsibility involves, it might be helpful to walk through a day or two and pay attention to what you are doing. How are you showing responsibility for your planet, your domain? What motivates you to do this? Where are you having trouble or not accepting a real responsibility?
Whatever I am doing, for whatever reason, and in what manner, and where I recognize my shortcomings -- those are things that probably are the readiest to hand as practical examples for my kids.
Even failure can be helpful -- if it is honest failure and not hypocrisy. My kids can see where I am struggling and I can talk to them about how I fell down in the past and what I did to improve.
Besides involving the kids in the daily life of our family and its upkeep, another way to teach responsibility is by stories -- not just stories of my own past, but stories from the news or from literature. For example, my boys are following the story of quarterback Cam Newton. My husband just pointed out to them that "character is gold" since Newton's character failings may end up costing him real money since some coaches are going to be reluctant to trust him. We see similar things going on in the local scene as talented players fail to succeed because failures in responsibility trip them up.
Of course, it's important to apply these stories and teachings to one's own life. It's easy to become judgmental of other peoples' flaws and indulgent towards one's own. For this reason, having a family culture that supports and encourages confession and improvement is an important thing.
I think "small things" is a useful term because it reminds me that responsibility isn't a huge thing that you can check off once a day. It's a bunch of little things. I don't like bunches of little things because they are many, and many confuses me. But on the other hand, they come readily to hand. Every moment I am doing.... SOMETHING. If I am thinking about my duty, what I should be doing this minute, this hour, this day -- I am thinking about my responsibility. ... which is basically my response to needs that face me, that I can do something about and should do something about. Every moment presents this anew. So it would be easy to simply look at my to-do list and come up with a lot of material to examine.
I'm going to try my own suggestion and keep a sort of log. I really want to remember that even failures and mistakes are fruitful opportunities, because recognizing them gives me a chance to grow.
I think I will make a resolution here to try to make my kids more aware of what needs to be done. This means bringing them into my planet, so to speak, which is sometimes difficult for me. The ultimate goal would be that my kids have a habit of scanning their environment for what needs to be done, discerning what they can appropriately do, and looking at their "toolbox" of competence to see if they CAN do it. It would also be important, then, to expand their toolbox so that there are lots of general things they know HOW to do.