A discussion of the validation of citizenship generally has already been linked, and at the time of writing this the censors' ability to remove individuals from the citizenship rolls has been mentioned, which doesn't really have anything to do with voter fraud. So far no attempt has actually been made to answer how Republican voting procedures ensured that only citizens were casting votes at elections and legislative comitia.

Simply put, the Romans did not validate citizenship at the vote. There is, for all practical purposes, no single indication in any of our texts that any attempt was made to determine the citizenship of voters. This is not an insignificant question. Plutarch describes at C. Gracc., 3.1 an assembly so large that people had to climb the houses adjacent to the Campus Martius to shout their votes down. Plutarch is confused, and his description is impossible since voting was not verbal, but even in his confused example Plutarch gives no indication that anybody bothered to confirm any of these voters' citizenship. The whole point of the episode is to show how many Italians--who were not yet citizens--came into the city for Gaius' election to the tribunate, and only a short time later the consul Fannius issued an edict expelling all non-citizens from the city during the vote on the leges Semproniae. Yet nowhere in any of our texts is any procedure confirming citizenship status before the assembly described with relation to this incident.

To quote Mouritsen: "there seems to have been no formal identity check or registration of voters." Mouritsen's argument, that the Saepta, where the magisterial elections took place, had a separate chamber where voting blocks waited before being called up, is shaky at best. Still, his statement here is absolutely true, even if there is no reason to suppose, as he does, that the marshaling of the voting centuries was some sort of unofficial check on citizenship: there is no sign in our texts of any attempt to check citizenship status before voting. Attempts by scholars--most notably Nicolet, who proposed that citizens received tokens that they could show when entering the voting spaces--to identify some sort of voter validation are without evidence and are not generally accepted.

The possibility of voter fraud existed in the Republican consciousness, however. I have already mentioned Fannius in 122. It's unclear then whether the fear was that the Italians would try to vote or whether they might introduce violence into the assembly, but there are a few other examples. Plutarch reports a story at Mar., 5.3-4 that a certain Cassius Sabaco, a senator, was brought to court because one of his slaves had been sighted inside the pens used to marshal the voting blocks. Sabaco's defense was that he had been thirsty and had called his slave to bring him a cup of water, whereupon his slave left the voting pen. The story indicates two things. In the first place, the Romans conceived of the possibility that a slave at the vote might be trying to cast a vote illegally. In the second, there appears in this instance to have been no identity check to enter the assembly. If there had been, Sabaco's slave surely would not have made it through, since he was readily identified once inside. Nor do other texts give any indication as to this possibility. If there was an identity check of some kind it must have happened as the voters were actually casting their tablets (the precise procedures of both the centuriate and tribal assemblies are debated, and the exact timing of this hypothetical check would have differed somewhat). But there is no indication of this in any of our texts.

This other thread which covers how citizenship was validated may be of interest to you.

Edit: pinging /u/Astrogator because he provided the linked answer and also because the automod told me to.