Yes, mail lockers, or milk delivery lockers.
I can see the argument that they were mailboxes. However, doesn’t this seem this a very inefficient way to deliver the mail?
This picture shows that there isn’t a keyhole for the postman to open all the mailboxes at one time. Would he really use some sort of skeleton key to open each box one by one?
Also, would the postman either walk up flight of stairs or take the elevator between floors? That would seem to also be very time consuming to deliver the mail to one 21 story building.
Package arrives from Montgomery Ward, Spiegel, or Allied Radio. Mail carrier knocks on your door, and if the maid isn’t in, puts the package in the box and puts the key in your letterbox.
UPS driver or paperboy could have done the same thing, except no access to letterbox, so maybe he left the key with the doorman or slipped it under your door.
My grandmother lived in an apartment in NYC that had lockers similar to that. I never knew what they were for.
They look to be the same size as the parcel lockers that accompany the community mailboxes in my 1980s-1990s built suburban neighborhood.
Anyhow, thinking if over further, I would think if they were mail lockers, there would be a locker devoted to outgoing mail which would have a slot for residents to put their stamped mail into. So…thinking milk delivery boxes is the most likely answer. Still, must have been terribly inefficient for the milkman to have to unlock each locker!
Not nearly as inefficient as stopping at individual houses and placing the milk in a box on each porch.
In China, they have arrangements that are pretty spot on, and they are for milk delivery.
When I delivered papers I would walk up and take the elevator down. It just takes too long to wait for the elevator between floors, but you really don’t want to walk all the way down from the top floor every time.
They look to be about the right size for tenants to store disembodied heads.
Wouldn’t it be easier to take the elevator to the top and take the stairs down? Or did you enjoy the workout? 
I actually found it easier to take stairs going up because I could take two steps at a time and keep the momentum going, and going down stairs is hell on the knees in the long run.
Open them all! Who knows what may be in them? Millions in unclaimed thousand-dollar bills! Action Comics #1! A treasure map with X marking the spot of a sealed, nitrogen-filled garage with a 1927 Bugatti in it!
Whether they were for milk delivery or mail delivery, the delivery guy only needs one master key to unlock all the boxes. He doesn’t need individual keys for them all.
The individual customers have keys that only unlock their own personal box and do not work on anyone else’s box.
At work I have one master key that will unlock every dorm room on campus. The 500 or so students have keys that only unlock their own rooms.