I would have read that story or had it read to me maybe several times by about 1961.
I remember it was in a large format book, which had soft covers or maybe had lost its hard covers. I think there were other stories in the book that I don't remember anything about, but I am not certain.
At least one illustration showed the family had a spaceship which might have looked like a flying saucer or a rocket ship, and it had a large transparent hemisphere covering the control room. And I remember an illustration of them seeing an asteroid with a human like face carved into it, presumably from the surface of a planet which had been shattered, producing the asteroids.
But I am not certain that I remember the story correctly.
Can anyone identify such a story?
It has been suggested I might remember scenes from Robert A. Heinlein's The Rolling Stones 1952. It was written years before I would have read that story or had it read to me, and it did feature a family traveling thorugh the asteroid belt. So I can believe that possibly a chapter from it or something might have been in some anthology of stories.
I note that The Rolling Stones was serialized in Boy's Life in Sept. to Dec. 1952 as "Tramp Space Ship", and there may have been illustrations.
And I might have read The Rolling Stones in the 1960s. I read all the Heinlein books in my local public library and the high school library. So possibly my memory might mix up a children's story and scenes from The Rolling Stones.
But so far nobody has mentioned a scene where a carving of a face is found on an asteroid.
I note that the idea that the asteroids were an exploded planet which once had a civilization is found in other Heinlein stories.
In Between Planets, 1951, it is mentioned that the "first" interplanetary empire in the solar system many millions of years ago was ruled by the fifth planet until it was destroyed.
In Space Cadet, 1948, it is discovered that a civilzation existed on the lost planet and the planet was desctroyed in an artificial explosion.
But I don't remember any Heinlein story where an asteroid was found with a carving of a human like face.
Added Nov. 15, 2021.
Another answer mentions the comic book Space Family Robinson starting in 1962. I bought them starting with the 2nd issue and later got a second hand copy of the first issue. The Robinsons were in the asteroid belt in the first few issues and later they were in interstellar space with no explanation. And I don't rmember them ever seeing a face on an asteroid.
Anyway, my memory of the story is that it was in a large format book, larger than a comic book, with soft covers or missing its covers. It was mostly text though it had at least one illustration.