Computers are less advanced, but the computer tech of 1968 is more than enough for a vigorous space program if physical expansion of living space is what the society wants. An ongoing babyboom would tend to focus interest in that direction I think.
You're fantasizing again. What doomed the US space program is not that white people stopped having as many babies as you'd have liked. It's that the economics of space exploration are marginal, and those of space colonization completely impossible. A lot of features of the space program could have gone another way, leading to more exploration today - stuff like asteroid mining and more space stations in low Earth orbit - but none of that would've dropped the cost of building a single habitable Moon dome (or Mars dome, etc.) to reasonable levels. And even then, the dome would be for roving teams of astronauts, and not settlers - think something like the Hab in The Martian, but bigger.
Space is not an ocean, and the Moon, Mars, etc. are not 17c New England, where you can drop a few hundred settlers in an area and they'll take care of themselves. 21c society is more complex, to the point that you need hundreds of thousands of settlers at a time to maintain living standards. The technical work required to maintain the habitation modules requires specialized skills as well.
That's on the supply side. On the demand side, why? English settlement of the US was not motivated by Lebensraum. In the 17c, England's population didn't grow. The motivation was profit: colonies in the South exported agricultural goods to the metropole, and colonies in New England exported manufactured goods to the colonies in the South. Businessmen could and did fund colonization. In space, not only are we talking multiple tens of trillions of dollars for a viable colony, but also there wouldn't be any profit in it to whoever was expected to pony up the money.