Context is as belows,
"One can mount its broad staircases or glide dreamily upward in its aërial elevators, attended by guides in brass buttons, with a serene joy that Alpine climbers have never attained."
Since this is written around the early 1900s, describing a hotel on Broadway, Manhattan, the term "elevator" on its own may have been relatively new, it is possible that aerial just makes the reader understand that it takes one "up in the air".
Existing or living or growing or operating in the air. "Aerial particles" , "Small aerial creatures such as butterflies" , "Aerial warfare" , "Aerial photography" , "Aerial cable cars"
100 years ago, perhaps it was OK, but I find the use of a diaresis in "aërial" very affected in the 21st century. It's OK in "naïf" for "naive" person, since that's an unnaturalized French word and the French use a diaresis, but there's no need for it in the English adjective, "naive," either. But "aerial" is an English word.
There's a book, " "Aërial navigation:a practical handbook on the construction of dirigible balloons, aërostats, aëroplanes, and aëromotors" by Frederick Walker 1902, obviously the diaeresis was very popular at that time.