I began thinking about this question today when I was visiting the Whitechapel gallery in Aldgate, east London.
I was there with my husband and a friend and we were in the Rothko in Britain exhibition which is upstairs in the part of the gallery that used to be a public library. It is a small room and there is one of his paintings on display plus quite a lot of his personal correspondence laid out in two cabinets that you can spend time reading. There are also photographs of his exhibition at the Whitechapel from 1961.
These are grainy black and white photos of people looking at the work and it includes a small boy, maybe about five years old, having a good look at a painting from its right hand edge plus another one of a younger child in a pushchair.
I often see small children being taken around galleries now but I don't recall ever being taken to any as a child. I do recall in the mid 60's being taken to Stonehenge by my parents and later on historic houses and areas of outstanding natural beauty.
I did a very small straw poll of my companions and they only began visiting galleries when they were old enough to decide for themselves what they wanted to do. In my case I was about 17 and I believe I saw the Pre-Raphaelites for the first time.