Can't discuss the Space Race without discussing why we had one in the first place. The things we were able to do were driven by political necessity more than anything else.
The Space Race was a by-product of the Cold War between the U.S. and the U.S.S.R. and was basically a contest to see who can claim superiority in space. For all the inspiring words about Human destiny and achievement, it really wasn't about that to the people controlling the purse strings. It was all about who could plant their flag on the Moon first and claim bragging rights to the entire world.
If the Space Race had continued, it might also have meant the continuation--and escalation--of the Cold War as its main driving force, IMO, which would have ultimately shaped its direction...
This is what I was thinking. This means today that space would be thoroughly weaponized. The likely scenario would be certainly no mission to Mars, probably no moonbases, but instead lasers, kinetic energy weapons, possibly some nukes, and so on, in addition to several sophisticated manned permanent space stations, at various altitudes including geosynchronous, orbiting the Earth today.
I always thought of the Apollo–Soyuz mission in 1975 as the symbolic, gentlemanly way of ending the space race in a draw. Both the U.S. and U.S.S.R. could claim victories: Sputnik, Gagarin, Armstrong. So, if the space race were to continue after the first several rounds, then let's assume there would have been no Apollo-Soyuz mission. For the space race to continue, the Soviets would have had to have done something dramatic and unexpected that scared the Americans, probably in space, and done it sometime in the early 70's.
For example, suppose we found out that they put a nuke in orbit. This would shape all aspects, not only of the U.S. space program, but of U.S. foreign policy. The space shuttle and U.S. space station projects would have been fully funded, instead of curtailed. As a side note, the Vietnam War would have continued until the United States toppled the government of North Vietnam. Nixon does not resign.
Now let's fast forward a few years in the real world to 1983. President Ronald Reagan announces SDI, threatening to escalate the arms race into space, in violation of existing treaties, including those prohibiting the placement of WMD's in space. That's our real world, and such things are not announced in a vacuum.
Now let's return to our imaginary world where the space race continued. We can practically guarantee that by now SDI would be a reality. Now eventually, macroeconomic opportunity cost must rear its head, on both sides. Therefore, a mission to Mars is out of the question. The U.S. space program becomes militarized, and NASA is perhaps absorbed into the Air Force.
In the unlikely event that moonbases exist, unlikely due to their expense and the trap of having to keep up with the other parts of the space race, then both sides have them. They are small experimental stations, probably there to test the feasibility of setting up automated weapons factories, or fuel mining, off the Earth. Then, the moon becomes one of the places where World War III could break out.