You mean what people could vote? States have always had the power to choose who can vote, but constitutional amendments have generally restricted the states' ability to bar certain classes from voting (XV, XIX, XXVI).

In Minor v. Happersett, the Supreme Court ruled that voting was not an inherent right of citizenship. Th Supreme Court has kind of backed away from this interpretation, but it remains true that not all citizens can vote. I'm somewhat curious how the court would rule on a law that banned people that met some random qualification from voting. As long as it wasn't along gender/race/age lines and didn't have a disparate impact, I wonder what the rationale to striking it down would be.

But I digress. Property requirements generally went away in the first half of the 19th century, non-whites got it with the passage of the 15th amendment in 1870, women got it in 1920. Native Americans were all granted citizenship with the passage of the Indian Citizenship Act in 1924. The mid-1960s saw the demise of many laws that de-facto restricted black voting and the 26th amendment lowered the voting age to 18.

But of course, states can always grant anyone the right to vote. Off the top of my head, they could probably grant people in China the right to vote, but don't quote me on that. Maybe there's federal legislation barring that, but even then, it could go to court.