What age should you be to start playing golf?
How old should you be to play any game?
The first lesson for any parent is that golf is a game. Games are about fun. Winning is the result of having way too much fun. If the child wants to play golf, then any age is OK to start. Just don't try to teach them yourself, don't force them and don't teach them on a golf course.
I happen to work at a golf course and interact with golfers of all ages. The common denominator for the golfers that are getting the most out of the game is that they enjoy it. They are having fun. Juniors that are accompanied by an adult are OK but they are better off playing when the course is next to empty because other golfers haven't got much tolerance for beginners of any age.
They need a putter, one iron (a 7 is good) and maybe a 3-wood that are all sized properly for them. They would be better off learning at a driving range on the practice green where they can putt and chip and learn the more important part of the game–the short game. Bobby Locke made the famous quote "You drive for show and putt for dough". And don't expect your child to be the next Tiger Woods. Your first priority is to ensure they are having fun.
An executive length course would be a good place to try in off peak hours. Once they show an interest, a junior program is a good approach at a local golf club but most parents use it as a kind of summer day care center where they just drop the kids off for the day. Golf can teach children the importance of respect, patience, joy of nature, manners and having fun. Forget the scorecard for at least 3 or 4 years. They will have trouble making proper contact with the ball. The forward tees should be where they start. They should never have been referred to as junior/ladies/mens/championship, etc. Beginner skiers start on the green beginner runs and so should all golfers and not just children.
If you see the magic in their eyes when they hit a straight shot and the sound of great joy, then you can see that further growth will be worth the effort for both you and the child. If you have to force anyone to play golf, there is no point. You should never have to force anyone to play a game. Otherwise that just becomes work. Children will grow up and have plenty of time to do work. Let them be children and play and have fun.
And while you are at you should learn how to be a child again and learn to play golf for fun. Maybe you shouldn't bother keeping score either and maybe you might enjoy the practice green, especially if you get to share that time with your child and you have fun together. Is there any better reason to play golf than to have fun? Probably not.