This strikes me as one of those things where the polled results vary drastically depending on how the question is asked.
The border situation (between whatever countries) is affected by globalization and communication, including the creation of the internet. When people realize how much better they can have it elsewhere (which is much easier to realize than it used to be), it creates more demand to move. But this results in a spike of demand that will normalize over time.
And this is because other effects of globalization have already started to normalize the differences between countries. For instance, it's not as cheap to offshore as it used to be, and that trend will continue. Human rights are increasing in other countries at a faster rate than they used to, just from the greater visibility of the rights people enjoy in other countries.
If we're all lucky, most of the normalization will be that other less advanced countries will improve, rather than other countries degrading - like hopefully we can all "grow" enough fast enough that it'll lift those boats, rather than sink the boats in the more societally/technologically advanced countries.
And that fear that "them getting ahead" might mean that "we will fall behind" is basically what drives the demand to "protect" our borders, even while generally people are in favor of everyone having a better life.
So I think it's totally possible for a majority of the country to be in favor of more open borders over time, while also being against entirely opening all our borders tomorrow.