I have no idea what age I was the first time I was hauled museum to the, it was NYC in the 50s, the 1950s there were museums with art, I'm sure I was there.
Since I donuts remember when I first went, there's no way I can offer an impression, I also don't know if it was an Impressionist exhibition.
I was probably .k. by it, pictures, colors it would have been exciting like opening a storybook with pictures/drawings in it.
Later when in school, during city stays we went to museums, I remember the sheer horror followed by delight when we ended up at the Metropolitan Museum of Modern Art instead of The Museum of Natural History (that may have been the 3rd grade with the teacher who had a new poem us learn each month and a piece of artwork).
I'm not a parent (sigh of relief).
An ideal age to start them there is unknown, children mature at different rates and have different interests. Debbie might love sculpture and hate Van Gogh
while Ralphie may love Van Gogh and detest sculpture. Each kid is different age both for and interests.
I've no problem with museums going after children and their parents to be patrons. The budget cuts make it so schools can't afford to educate these children about art. Same as for music, if Lincoln Center wants to go after families to entice children, then it's good to me. In grade school at one point when back in the city, I was given a ticket to a local amatuer production of The Barber of Seville, at the local college, it introduced me to live theater, I was 10 or 11 at the time. So, yes, museums and other arts segments (theater, music, dance, literature [library]) should come after families and even offer discount rates for them.
General children in ate reasonably well behaved in museums and don't get excessively loud. My theory is the high ceilings, weird lights and marble floors freak them out into a peculiar haze. Take them to storefront galleries like the ones in Chelsea they are not as well behaved. Let the museums after the children and parents come, teach the little ones. tools, Give them the and they can then pass that rich history on to the next generation sharing what touched them most.