Does the electorate have a say? Do they vote in referenda to change state constitutions?
In some states, the constitution can be changed by popular vote. Texas and Oklahoma are notable for having to make every little adjustment by constitutional amendment; it's pretty messy, actually!
However, the state legislatures (bigger than your city council, but smaller than parliament :-), have to vote on amendments to the national constitution. And unfortunately, in terms of the "born citizen" issue, a state can't vote to reject the children of foreigners as citizens, because the national law overrides.