I don't have have any kids, but I can give you my impression from the instructor side of things.
At the mountain I taught at, the policy was the kids had to be at least 5 or 6 before they were allowed to take snowboard classes. This is due to the development of improved muscle mechanics/balance that happens a few years after the little one starts to walk.
3 years old was the limit for skiing, but that's because all the child needs to do is stand comfortably. Also with skiing, they can use that little bungie thing that holds the ski tips together to prevent runaway 3 yr olds all over the hill. Nothing like this is available with snowboarding, so the child needs to have better muscle mechanics and balance to be able to make turns/stop.
I would suggest starting your child in ski classes, just to get them some slope time at a young age. This will get them used to the "uncomfortable" gear that many kids get annoyed with. It also lets them get used to sliding on the snow, and decreases their fear when it comes to getting down the hill. It also gives you the opportunity to ride...
Then, when they hit 5 or so, throw 'em on a board and they will be teaching YOU how to ride in only a few years... I'm always amazed when I see a 8 year old ripping down an intermediate run, throwing 180s and trying to grind on every little tree branch he can find.
In the meantime, grab one of those little, plastic sled style boards (basically a plastic sled shaped like a snowboard, no bindings, just sticky pads to stand on and a rope to grab) and let your kid slide around on it while sitting. Like any adventurous child, they will probably try to stand up on it at some point... --> Pelican Space Scooter Snowboard : Target
kinda like the above, only with a rope instead of handles.