This is a question I hear a lot. And obviously it depends a great deal on the child as an individual, how mature she is, what are her physical capabilities, etc. But I can offer some basic guidelines.
I believe learning music is like learning a language. We don’t expect babies to speak in their first year. However, we continually expose them to human speech in the environment, so they begin to absorb the rhythms and sounds of language. So too, with music. By listening to music, played by a parent or an older sibling, or a recording, they begin to form notions of musical rhythm and melody.
At age 3 or 4, most children will sing and clap along with their favourite Disney movie songs. This is also great development for learning musical ability.
In my opinion, most children do not have the necessary combination of balance, coordination and motor skills to begin learning to play the violin until around age 5 or 6. At this point, they can learn to hold the violin and bow correctly, and can commence actually playing simple melodies within a few lessons. Older children will progress somewhat more quickly through the early lessons, but the process remains the same.
There is no upper age limit for learning to play an instrument. When a lifelong love of music is instilled at a young age, you might find yourself at 40 wanting to learn to play a new instrument.
Check out this great interactive by the Toronto Symphony Orchestra on rhythm and melody: