I've been writing to pay the bills since my children were very young. As long as they have been aware, they have had the experience of having people comment to them - and to me in their presence - about books or articles or essays or blog posts that I've written.

(Me, at work, blogging from Bonnaroo, 2006)
Mostly, that's a pleasant experience for them because 99.9% of the time, the people who take the time to talk to them or around them about their mother's published writing are doing so because they have something positive to say - stuff like "I enjoyed that article your mom wrote for the newspaper last week," or "I like your mother's blog," or "It must be fun having a mother who is a writer."
But occasionally, very occasionally, someone will say something disagreeable to one of them about something I've written. It happened recently when a teacher told one of my kids that he hated a recent blog post of mine on a controversial subject. The child in question mentioned the teacher's comment to me, and wanted me to explain what I had written (the kids don't keep up with every single thing I publish, although they do read my personal blog pretty regularly). So, I did. And that was that.
Then, yesterday, I had a lengthy interview published in the newspaper with a very controversial and not-well-liked figure in a rather infamous local murder case. Because this character is so unpopular in our city, and because some folks seemed to believe that the fact that I interviewed her meant that I was somehow absolving her of any culpability, the online comments following the story have been unusually vitriolic toward me, with one person suggesting that I "rot in hell."
I had in passing mentioned to my children that I was working on this story, but I hadn't really wanted to get into it too deeply with them due to the adult nature of the topic (adultery, murder, etc). But fearing that they might see the story on the front page of the newspaper, or somehow stumble across the comments following the online version of the story, or even hear from adults who had read the story, I felt like I should discuss the matter with them.
So, I explained to them that the interview had been published, and that because a lot of people believe that this woman, the one I interviewed, was a very bad person, some folks around town felt that I shouldn't have interviewed her. I told them that there were some unpleasant things being said about me in the comments attached to the story. And once again, as I do periodically, I asked them what it's like for them to have a mother who writes under a byline that also happens to be their last name. We had a good discussion, and they shared the pros and cons of how my work affects them.
As I explained to them, I totally get what that experience is like for them because when I was a child, my mother was a newspaper reporter and editor, and my father was a TV reporter. Sometimes that was a fun thing for my brother and sister and me, but sometimes it was a real pain in the ass to listen to angry local citizens who decided they would explain their objections to my mother's journalistic judgment directly to 13-year-old Katie instead of just writing a letter to the editor.
My sister reminded me yesterday of one time when we were in middle school, and a teacher used our mother as an example of "yellow journalism" when he was lecturing our social studies class on the importance of a free press. And I well remember the time a friend's father angrily informed me that he would no longer be watching the television channel on which my Dad appeared each night because of my father's "obvious bias" toward one candidate over another in an upcoming election. This happened at a sleepover, and I was mortified beyond belief.
On that particular occasion, I remember wishing that my parents did something more low profile for a living, but most of the time, I found it wonderful to have two parents who were constantly communicating with and about the world in which I lived. Having two parents who actively and publicly engaged in civic discourse and social debates helped my siblings and me develop our own views and learn how to express them with confidence.
I like to think that this will be the case for my own children - that they will be more confident and opinionated and willing to speak up for things that matter to them because they see their mother doing it on the job. They are already pretty opinionated kids, and I encourage that. I want them to be engaged and active members of the community, whether that's as journalists or public servants or bloggers, or even just as citizens with a point of view.
In the meantime, though, I'll bet sometimes they wish they had a mom who did something "normal" like sell real estate or teach school.