My idea of a "good family portrait" has changed.
At one point, I wanted them to look at the camera and smile, all dressed up in cute outfits that were neither slobbery nor wrinkled.
However, I love the pictures we have of our family. Our "good family portraits" include one where my kids are dying laughing (laughing hard enough to have closed eyes), one with my then-2-year-old having a ball attempting to escape my husband's lap, and one where you can tell my oldest child is mid-sentence.
They're perfect. :-)
I would suggest, have them in nice but comfortable clothing, usually dark works well on everybody. For a very young child, sometimes, having a teddy bear or something in the picture is better than having a screaming child. If you're child is impossibly attached to a dirty blanket or whatever, let them sit on it. If you have a daughter who simply MUST have the Cabbage Patch Doll she carries with her everywhere, put her little baby behind someone else in the picture, or anywhere where it'll be out of the range of the camera. As in, if your daughter's hands won't be showing in the picture anyway, having someone hold her baby right out of range so that she can hold the baby's hand would make virtually no negative difference, but could keep her from having a meltdown. Same goes for holding Mommy's hand. Make an appointment with a smaller, slightly more expensive place if you can. They will be able to accommodate you for a longer period of time, will have people usually with more experience (they have higher standards) and their wait time will probably be shorter. Also, schedule your portraits on a weekday, instead of Friday, Saturday or Sunday. One of the great small-child-portrait-killers is the long wait time. They get there and are semi-ready for their pictures, then they're stuck in a room with too many people and too few chairs, and not enough to do, and they can't get dirty or mess up their hair or anything... It's awful.
Specify when you make your appointment that you have small children, and ask for whoever is best with small children. What can really help is if you go to the same person every time you get portraits. The kids know the person, so they don't have to try to build a repoire (spelling that wrong). However, if you go with someone who is really good with kids (not fake good with kids, trying too hard.... they won't take him seriously, and if you're in the pictures, too, it'll show in your eyes that some FREAKZOID is taking your picture). Part of being "good with kids" in this instance is to be quick and efficient. They will not happily sit and smile for one shot after the other. Instinct is REALLY important in someone photographing little kids.
A lot of children are scared of the places where they're placed on a table to pose for a portrait. If you think this could be the case, find a studio that has them on the floor instead. Many of your mid-to-upper class studios have these, and tables are more likely to be found in Walmarts or other discount studios (great for babies, or otherwise kids who don't mind the table, but it's true, you'll probably be using a table pose at Walmart) or some of the JCPenney or Sears portrait studios.
Make sure your children are rested and fed and have been to the bathroom. Bring a quiet toy or favorite book for any wait times. Be upbeat and realistic about what you can expect (expecting what your children can't give is not going to make them more likely to give it).
The main thing is to adjust what you may consider a "good family portrait". Think of it this way: any parent knows that sometmes, the "imperfections" in our children are the most endearing. Trying to get a good portrait is like trying to capture a personality in one instant.