It depends on what you mean by proper training.
I used to teach an introductory karate class which was for younger children (5-7 year olds) which was a step to the main class. It was more about learning the very basics and dojo etiquette with lots of fun games thrown in to keep a varied lesson.
Now, if I had a young kid and wanted them to learn MA I definitely would not take them to karate. I have a few reasons but chief among them is that I don't believe it is a good idea to teach kids to punch and kick.
I think a good judo class, or something similar, would be more engaging and more useful to a child.
As for age...I think it is dependent on the style.
For judo, if there is a specialist kiddies class, I think 5 is a fair age. But the emphasis should be on fun.
In a mixed class this age would go up.