On 6 July 2005 I climbed the Alpspitze with my two daughters, Samantha (13) and Jordan (11).
There is a lot of loose rock on the mountain and at the time we climbed there was melting snow and ice that would periodically cascade small rocks and ice across sections of the route. These sections are obvious when you are up there, and I highly recommend a helmet because of this. My daughter Jordan caught a thumb sized rock with her helmet on this climb that would have knocked her silly if she wasn't wearing it.
If you want to take your children up the Alpspitze, get them out and up an easy route at your local crag. If they can handle the heights there, the Alpspitze, though intimidating in spots will be easy by comparison. It is not the mountain to expose them to for their first time out, but is a great introduction to traditional rock climbing. It has no real crux, just the perception of serious exposure. Adults with a lot of climbing experience will enjoy the view and seeing their kids respond to the challenge.
There is really no water on the mountain, so carry it up with you. Catching the first tram up the mountain will give you plenty of time to make it up and back. We poked around on the way up and down, and although my girls were tired at the end, they were not completely wiped out by this climb.
I highly recommend this climb for families with active, energetic children that are old enough to be safe on their own, and willing to be challenged. If you have any doubts about their reaction to exposure to heights there are plenty of other easier routes in the area. This is not a route you really want to turn around and climb down although it can be done. As far as my girls are concerned, they still talk about it, and looking up at the peak from the valley floor gives them a great sense of pride and accomplishment in knowing that they climbed the most beautiful peak in the valley.
See the Via Ferrata route page for more information on what you will need for a successful climb of this route.