The situation you describe, where the laws would be selected to make obtaining a photo ID (and thus voting) difficult for one group over another, would amount to a poll tax — which has already been found unconstitutional in the United States.

However, every US state I'm aware of (if you know of exceptions, please let me know) has some form of photo ID you can obtain for free. Usually it's in the same format as a driver's license, and obtained from the same place (the Department of Motor Vehicles, Department of Land Transportation, the name varies). There's a fee to obtain a driver's license, but a photo ID card (in the same format as a driver's license but marked "NOT LEGAL FOR DRIVING" or similar wording) can be obtained for free, by going through the same process as getting a driver's license (bring something to prove your identity, get your photo taken, wait for the card to be printed) but without taking a driving-skills test.

And before someone asks "what about the people who can't prove their identity"? Well, I can actually give an anecdote. I know someone who ended up in that situation: away from home for college, couldn't get her birth certificate or anything else, and needing to replace her passport that was lost. (She is American but had grown up overseas because of her parents' job, hence why she had a passport but no driver's license when she went off to college). She had quite a time of it at first, since every "prove your identity" requirement circled back around to another form of ID. But she was eventually able to get a fishing license just by swearing under penalty of perjury that she was indeed (name). With that in hand, she got another form of ID (I think a library card, though there my memory is iffy), then with two forms of ID she could get something else, and then eventually she was able to get that non-driver's ID card... and then was able to prove who she was to get her lost passport reissued.

That all happened nearly 30 years ago so I can't swear to the details. But the point is, photo ID cards are widely available, and it's normal to be asked for one. So as long as the law specifies that acceptable forms of photo ID include X, Y, and Z (where X, Y and Z are forms that most people would already have, and that are normal and in common use), and as long as at least one of those forms can be obtained without a fee (which as I said, is true in all U.S. states that I'm aware of, please mention any exceptions you know about because I'd like to know) then it will pass Constitutional muster.