So many ways to respond to this question...

My opinion is that parents create the environment and provide the opportunities and it is up to the child to develop a passion or not. Have music present and accessible from infancy onward. Have toys that allow for motor skill development and musical (subjective term at that age, of course) expression. Play your instrument(s) and foster an appreciation of music.

I am a believer in the Suzuki method, and like that it brings attention to having respect for the instrument as well as introducing instruction in music. Violin is a good starter instrument, specifically with Suzuki. When my son had the basics of reading sheet music and fundamentals, he transitioned to piano and began receiving instruction. It is important that the instructor knows how to relate to children. Make it fun and make it challenging. My two kids are 14 and 11, and the older continues to take piano instruction (8 years and running) and he chose to take up percussion in school band. The younger plays drums and just started back in with violin after a 4 year break with it, upon his own idea.

I took piano instruction at six years old and liked but didn't love it. It did teach me music fundamentals and a start on reading sheet, and after dabbling in trumpet I took up guitar at fifteen. The one thing I wish I had focused on, and have tried with no success to push my kids into, is voice training and singing. In hindsight, I would have put my kids into chorus early.

That's my experience. I'm interested in what others have to say.

I agree with Peter D. After a certain age it really is up to the child to develop as a musician. 

When my son was little we made sure he had piano lessons for a couple of years, and from that he gained a good foundation of basic music theory and reading. I made sure he was exposed to a ton of music, and had access to a variety of instruments. Now at 14, he studies the drums formally, but spends at least as much time playing the piano. He plays saxophone in the high school band and piano in the high school jazz band. He also has a better grasp of the guitar and bass than most guitarists his age, and is teaching himself to play the flute and trombone. We do insist that he practice the drums because that's what he has committed to studying, but leave the rest to him. I think if we insisted that he practice all this stuff he would see it as work and refuse to do most of it, but because it's his own thing he is free to explore and play with it. We do offer to get him lessons in any of his side instruments, and occasionally he decides to take a lesson or two, but it's his choice. 

I think the number one thing is to share your passion for music with your child.  Children are excited about what you are excited about when they are young.  I suggest using this to your advantage.  I did with my daughter who is now almost 6 years old and she loves to play piano and drums with me.

Here is a video I made for the parents of my students to help them grow a musical home.

https://www.youtube.com/watch?v=zC1O3B7IQU8