Common sense would say that Constitutional rights apply only to citizens. Some protections should extend to non citizens here legally. It should go without saying that only citizens have the right to vote - especially when you consider that people owning multiple properties can only vote where they live ... allegedly anyway. With respect to Indians and proof of citizenship, I'd turn a common phrase to the flip side and say "No representation without taxation."
There is an alternative. The state could simply declare all voter registration invalid until the registered voter has tendered proof of citizenship - a one time deal instead of proof at the polls. That also would give the state a clear path to cleaning up the voter rolls - prove you are alive, a citizen, and resident.