These are what I call the Letterman Payphones. The first picture shows a group of payphones as they looked circa 1999. The next photo, from roughly 10 years later, shows that those 3 payphones were replaced by 2, with an advertising-laced enclosure placed over them.
I call these the Letterman Payphones because Dave Letterman used to call these payphones from his desk at the nearby Ed Sullivan Theater. He called other New York City payphones too but he called these phones on a number of occasions when I happened to watch the show. His intent always seemed to be random, and depending on who answered and what kind of conversation ensued Dave might invite the person into the studio, ushering the bewildered passer-by onto stage as hundreds of adoring fans cheered wildly.
Like a lot of Dave's shticks it was not always easy to tell if it was planned. I used to call these payphones myself and all I can say is that when I called the people who picked up had things to say that were not suitable for broadcast television. The first words spoken clearly indicated that the person who answered expected this payphone call would lead to some kind of encounter. Letterman, through perseverance or planning I do not know, seemed to often get all-America types of tourists and everyday folk.
The last time I saw this routine on the show it seemed obvious that the person who answered the phone was expecting the call. Maybe Dave himself planned it or maybe the person (who seemed to be an actor) got word from sources that a call was coming to that payphone. Whatever the case it was not random and not very funny, and I seem to remember hearing a bit of dismay in Dave's voice when it seemed obvious to him that this person expected the call.
Does Dave call payphones any more? I do not know. I watch the show once in a while -- a friend and I attended a taping live at the Ed Sullivan Theater earlier this year -- but I have not spotted a payphone bit in a long time. Payphones are mostly programmed to reject incoming calls, and unless Worldwide Pants (the show's production company) is able to circumvent that by arrangement with Verizon or individual payphone owners then it seems like the Letterman payphone skits of yore are no longer possible, at least through these and other midtown Manhattan payphones.
The numbers for these phones are (212) 246-9175 and (212) 974-9778. These phones actually can accept incoming calls, though it is unlikely that a random passer-by would pick up. The phones ring very faintly, and they drop to their internal modem after the first ring. If someone at this phone is expecting your call and standing right at the phone then they should be able to hear it ring once and pick up before the modem kicks in.