I gave one of my kidney's to my brother. I am married, and am the sole provider for my wife and 2 children. My boys are 4 and 6 years old. My wife and I struggled with the idea of kidney donation. My first instinct was, of course I will donate my kidney to my brother! My wife's first instinct was, I hardly know this guy, and I'm going to put the life of my children's father on the line for him? We struggled greatly with our decision. Notice I don't say MY decision, as it clearly was not a decision I could make on my own. I don't know how to describe to you the real agony we went through struggling with this. I didn't realize it completely at the time, but I think my wife felt like she couldn't really say no. So, even though she was AMAZINGLY supportive, especially during my recovery, I don't think she was ever totally on board. I am sure, if I asked her now, she would say NO WAY! (and my surgery and recovery have gone very well)!
Sherri correctly points out that there are REAL risks. I have not studied liver donation, but the amount of research that has been done on the long term effects of kidney donation is laughable. I assume it is the same for liver donation, but perhaps someone else can chime in about that.
All this is to say that your feelings are totally valid, and it only makes sense that you would be looking out for the best interests of your children. I have two pieces of advice for you: Please make sure your husband has all of the life insurance you could ever want BEFORE he does a single test. Secondly, get a second opinion about your husbands suitability to be a donor, don't rely on the transplant center alone.