In reply to cheek to the rock:
My daughter started climbing at 3. The only issue we had was finding a harnss that would fit her, plus a wall that would let her in - several have a lower age limit, although this can be hard to work out as their lower limits are often different for instructed courses compared with taking your own family.
We only took her because she asked to go, and it is always on her terms. I definitely don't get any climbing done when I take her, just spotting and belaying. I thought her attention span would be short, but she often doesn't want to leave for several hours (with snack breaks). It helps that our local walls have quite a number of low grade routes and a dedicated kids' bouldering area.
It is definitely better to go at quiet times as young kids are not as aware of other climbers as they might be.
Post edited at 08:14