my daughter is too young to think about this yet (only 6mo old) but in a few more years we will be putting her on some sort of shred stick whether its skis or a board I'm not sure yet. I think the "skis first" is probably changing with the times, as more people snowboard there are a lot more people who know it and know how to teach it to younger kids. although skiing seems somewhat more intuitive, there are plenty of ways for an uncoordinated kiddo to screw it up on skis (crossing tips, wobbly, etc.) that wouldn't happen on a board and vice versa.
Personally, as much as I'd like to start her sooner rather than later, I think my rule will be: she can't go on the hill until she stops shitting her pants (and preferably until she knows how to wipe her own ass). Keeping your kids clean is enough of a hassle without the added trouble of boots and snowpants and baselayers getting in the way.