A nice illustration of "Dead Man's Curve" is on the cover of Harper's Weekly, March 27, 1897.
In the golden nineties
by Henry Collins Brown
Hastings-on-Hudson: Valentine's Manual
1928, c 1927
Pg. 58: The cable-car innovation on Broadway (afterward changed to underground trolley) also brought a new local term into the city's nomenclature--"Dead Man's Curve." This was the curve in the line at Broadway and 14th Street which occasioned the city's first experience in dodging fast-moving vehicles coming around corners. The old time horse-cars were no source of peril of pedestrians at this point, but the cable-cars were compelled to make the turn at an accelerated speed that was the cause of many casualties and the consequent (Pg. 59--ed.) sinister appellation. Extra policemen were stationed at the curve, and their shouts to the unwary mingled with the hideous clanging of the motorman's bell, and the shrieks of frightened women made the crossing a scene of pandemonium for a long time, until it was recognized as a danger point, and traffic regulations instituted.
Grant Barrett (Oxford Dictionaries) posted the following to the Wordorigins.org site:
The earliest cite I could quickly find is from the 7 Jan. 1888 Portsmouth (Ohio) Times, p.1: "'Dead Man's Curve' is a piece of track on the main line of the C.W. and B. between Madisonville and Madeira. Seven persons were run down and killed there during the past year."
The earliest I could find in reference to NYC was from the 3 Sept. 1892, New York Times, in a page one article datelined St. Louis, Sept. 2, talking about a foiled train robbery: "At a point called Dead Man's Curve the freight was slowed up, the piece of track being very dangerous."
An article from the Times dated 20 Jan. 1897 supports the book linked above: "One thing only, in the estimation of New-Yorkers generally, is more fatal to human life than knockout drops, the Brooklyn trolley, or Seeley's diner, and that is 'dead man's curve' on the Broadway cable where the track makes two turns close together in going around the corner of Union Square at Fourteenth street. The cars make the turns at the highest rate of speed of which the cable is capable, and the unsuspecting countryman from the outlying parts of Great New York may see a cable car point away from him at a 90Â° one second and be surprised, ambushed, and slain by it the next."