I don't think that a spacex type company could have survived during the 1980's and I totally doubt that they could have achieved success. The 1980's were a decade of change, the major change coming from ready accessability to the computer. Obviously, we all know what happened in the 1990's but what people don't understand is the complications of this. This increase in productivity, particularily in the high science and math related jobs, is what allowed SpaceX's existance. They had to be able to lower the sunk costs (development costs) of a rocket. In particular, the rocket motor that is probably one of the most complicated things I have ever seen. In the 1980's, this meant one thing...run the number...build it...test it. Private companies did not have the shear resources (or computing strength) to run complicated simulations. Thus the costs of a private company developing a rocket by themselves (spacex style) were WAY to high. Also, there was less traffic needing to go to space. There wasn't the potential revenue stream there is now. Basically, if rocketry went private in 1980 there wouldn't be a space program.
Let me explain. When making business decisions there is one factor that ALWAYS comes up eventually. Yes, there are a million assumptions that go into it but thats what analysts are for. This factor is NPV (Net Present Value). This number needs to be sufficiently high in order to partake in the business (how high is sufficient is up to your utility). Space rocketry has been a negative NPV project for years and has just recently increased to the point where investors are willing to take on the risks. There was no profitability in the 1980's for a private company...if there was it would have been developed.
But aside from the reasons there wasn't a private company doing it, I'll answer your question. I think we would have a sizeable amount of rovers on multiple bodies. I also believe we would have a relatively robust infrastructure in space (tugs, fuel depots, even habitats). I believe that if there had been Spacex in the 80's, it would have spurred on the development of a Bigelow. This utter lowering of costs would have caused NASA to completely rethink its direction. Because they could put up large habitats in space, they would have started to develop these ideas. I believe that we would have an ISS that is 10times the size of the current one. I also think it would have profitable purposes (satelite garage anyone). But this is all wishful thinking...alas we are stuck with what we have now. But man does the future look bright