Truth be told, it has very little to do with actual voting and everything to do with money.
Right now states issue IDs and as long as those IDs are not used to vote or exercise any rights, they can charge money for them (this is one reason it's constantly emphasized that driving is a privilege, not a right). However if your driver's license is used to vote, then charging a fee for issuing a driver's license means it's a tax on voting, a tax on exercising a constitutional right, and the Supreme Court has repeatedly shown a willingness to come down like the the fist of an angry god on anyone doing that. They would have to quit charging for state IDs and driver's licenses and that means tens of millions disappear from state budgets.
But, says you, why would a national voter ID cause that problem? The states could still charge for their licenses and all would be well. Theoretically this is true but in reality, we would start using that national voter ID for other ID purposes. We already use Social Security numbers for all sorts of IDing purposes that it shouldn't be simply because it's a national ID and convenient, despite not even being a photo ID. With a more secure, more valid ID, more and more ID purposes would be shifted to that ID and fewer people would require a (paid for!) state ID. Now driving would still require the state ID and in Republican states, that'd be fine. But New York? There's twice as many citizens as cars and probably less than a quarter drive regularly. They wouldn't need to pay money to the state for their ID anymore if a free national ID served for all their purposes except driving. Los Angeles? San Francisco? Not as car-free as New York but still easier to take the bus. The big cities would all lose fat stacks of cash because fewer people drive there.
Much of the gnashing of teeth about voter suppression is really a smokescreen for the real issue of national IDs: Making sure fees for driver's licenses continue to flow into the coffers of large cities.
