Nobody is pushing for an "open border" and the absence of a voter ID leading to fraud is just a dumb conspiracy theory promoted by people that are ignorant of how elections actually work.
All voters need to be registered in the first place at which point they verify your identify, your ability to legally cast a ballot, and your address (falsely registering is a felony in most jurisdictions). On voting day, you show up to the correct precinct and tell the administrator your name, they typically verify that your address is correct and then you can cast a ballot. Casting a fraudulent ballot is also a felony.
So the theory is what? That all of these illegal immigrants are going to register to vote? They wouldn't be allowed to register as non-citizens and falsely attesting is a crime. They would show up on election day and cast a ballot under their own name? They're not registered, so their votes wouldn't be counted. That they're going to imitate an actual voter on election day? Instant felony which is easily caught if the real voter shows up at any point to cast their own ballot. That they're going to intercept the mail-in ballots somehow? Again, when real voters figure out their ballots are missing but votes are recorded in their name, the fakes would be immediately found out.
When the states advocating for voting IDs have a long history of race-based voter suppression, analysis shows the ID mandates have race-based impacts that would suppress votes, and there isn't an actual "attack surface" that would be solved with voting IDs, it's clear it's just a transparent attempt to suppress votes.