Where I live (NYC), the minimum wage is $15 an hour, which is certainly better than the federal minimum, but is still basically unlivable in New York. Cheap rent here is on the order of $1,200 a month (and that's for a roach-infested, lead-painted mess of an apartment), not even counting any other expenses.
It's expensive enough that I think it prices out a lot of low-wage workers, so I think the answer is "yes", the wages are still an issue, at least in some places.
You must either really mean Manhattan or you must not have ever looked, because $1200 isn't the bottom for a bedroom. I make software eng. salary and lived in hip parts of Bushwick for 4-5 of the past 6 or so years and paid 875-1100 for a bedroom, and Bushwick is a 'cooler' part of Brooklyn. 30 min to 14th St. Union square in Manhattan on a subway. You can look on Streeteasy right now in Bushwick and see 4 bedroom apartments that net sub-1000 dollar bedrooms. These aren't roach infested, lead paint units, a fair bit will be shoebox sized rooms but modern renovations. If you were to look in an actually poor neighborhood in an unsafe area, or were to push your commuting distance to 45-60 minutes parts of Brooklyn and Queens, you could find cheaper.
But it's still true that the poor are being priced out and doing it with a family (with any semblance of privacy / not living 4 people in a bedroom) would be horrid.
I did mean Manhattan, sorry, I should have clarified.
Even still though, let's assume you're paying somewhere around $900/month, that's a pretty tight budget for someone making $15/hour, especially if the employer isn't giving you enough shifts to qualify for "full time". My brother in law worked at Taco Bell when he was living in NYC, and they never gave him more than 29 hours, because they were afraid of having to pay real benefits. Fortunately for him he was able to stay with us and I have a yuppie engineering job and a spare bedroom, but I can't imagine what people who have kids working at $15/hour actually do.
If you can get by in a nice part of Brooklyn on a minimum wage salary - that seems not terrible.
Do we really think there was a time in the past when a head of household worked at McDonald's with a stay at home partner, two kids, and car in the garage with a white picket fence in East Hampton?
No, 2 of the apartments I lived in there had central AC and heating, one had a beautiful kitchen island and skylight, the cheaper units were 3-5 bedroom units though.
Rents have increased 20-25% in the past 1-2 years for sure, but it's still not $1200 as a minimum.
It's expensive enough that I think it prices out a lot of low-wage workers, so I think the answer is "yes", the wages are still an issue, at least in some places.