The 14th Amendment made it so every person born in the US, slave or not, would be granted full US Citizenship. How did it work prior to then? Was it simply such an arduous journey to get to the US prior to the 1860s that it was never really an issue? Was it simply a case of declaring your residency with the state so you could vote in state and national elections? There were obviously no passports back then and you could technically travel just about anywhere as long as the country you were entering was cool with it.
You have to pull apart the issues of immigration, citizenship and voting a little bit. The 14th amendment had nothing to do with how people crossed borders, it described who could be naturalized as a citizen. You are right, before and after the 14th amendment you could get into the US easily, all you had to do was show up. But that didn't necessarily make you a citizen. You could live here your whole life and not be a citizen. It wasn't about who had documentation. So we were happy to have Chinese laborers come into the country and work and live here for a long time, but we would not grant them citizenship. The idea was that they would leave eventually, but there was no formal bureaucracy overseeing that.
And citizenship did not mean you had the right to vote, before or after the 14th amendment. Only white men were granted the right to vote before the 14th amendment, and in former slave states black folks right to vote was taken away after Reconstruction. Women were not granted the right to vote until 1920.
Its not until the 1920's that we start guarding the border. The 1921 Emergency Quota Act was the first time we limited who could enter the country based on the place they came from. This was to keep Jewish people fleeing from Europe from coming into the US. The Act had the effect of limiting the number of people who could come from eastern and southern Europe into the US.