For voters it was likely then as simple as it is today, just being present. All an immigrant has to do at the federal level today is sign a statement under penalty of perjury that they are a citizen. No proof is required.
With that, many (dem?) states claim if an immigrant has satisfied federal requirements "who are we to disagree".
Hell of a way to run a country. Yes, there have been suits in the past and pending even today.
Yes, hell of a way to ruin a country.