West Broadway (known as Laurens Street in the 1800s) was called “Rotten Row” since at least 1840. It was a place of dilapidated houses, crime and prostitution.
 
 
Wikipedia: West Broadway
West Broadway, not to be confused with Broadway, is a north-south street in the New York City borough of Manhattan separated into two parts by a park. The northern part begins at TriBeCa Park, near the intersection of Sixth Avenue, Walker Street and Beach Street in TriBeCa. It runs northbound as a one-way street past Canal Street and becomes two-way at the intersection with Grand Street one block further north. West Broadway then operates as a main north-south thoroughfare through SoHo until its northern end at Houston Street, on the border between SoHo and Greenwich Village. North of Houston Street, LaGuardia Place continues along the same path.
 
The southern part of West Broadway runs southbound from TriBeCa Park through the TriBeCa neighborhood, ending at Park Place. Prior to the September 11, 2001 attacks, West Broadway continued southward into the World Trade Center site, ending at Vesey Street. It was once considered “Rotten Row”.
 
17 February 1840, New-York (NY) Commercial Advertiser, pg. 2, col. 3:
FIRE.—Yesterday morning, about 3 o’clock, a fire was discovered in a cluster of wooden buildings, known as “Rotten Row,” Laurens street.
   
Google Books
The Women of New York:
Or, Social Life in the Great City
By George Ellington (pseud.)
New York, NY: The New York Book Company
1870
Pg. 571:
The famous Rotten Row, in the neighborhood of Grand street, consists of bad, dilapidated houses.
 
Google Books
The Dangerous Classes of New York:
And Twenty Years’ Work Among Them
By Charles Loring Brace
New York, NY: Wynkoop & Hallenbeck
1872
Pg. 94:
... the notorious rogues’ den in Laurens Street — “Rotten Row” — where, it was said, no drove of animals could pass by and keep its numbers intact; and, farther above, the community of young garroters and burglars around ...
 
OCLC WorldCat record
Gotham’s Greater Rotten Row
Author: Municipal Improvement Association, New York.
Publisher: [New York] Municipal Improvement Assoc., 1890.
Edition/Format:   Book : English
 
Google Books
The Gangs of New York:
An Informal History of the Underworld
By Herbert Asbury
New York, NY: A.A. Knopf
1928
Pg. 243:
But these slums were equalled if not surpassed by such places as Rotten Row in Laurens street, Poverty Lane and Misery Row in Ninth and Tenth avenues, and Dutch Hill, a group of shanties in East Fortieth street near the East River.
 
Google Books
Low Life:
Lures and Snares of Old New York
By Luc Sante
New York, NY: Farrar, Straus Giroux
1991
Pg. 27:
They might go to the Rag Pickers’ Den, between Pitt and Willett Streets, inhabited by Germans and ravaged by cholera; to Rotten Row, on Laurens Street (now West Broadway), where, in the words of one contemporary, “no drove of animals could pass by and keep its numbers intact”; ...
 
Ephemeral New York
March 16, 2011
West Broadway: once a slum called “Rotten Row”
Luxe boutiques and galleries have lined Soho’s West Broadway for decades.
 
Which is why it’s so hard to imagine that in the 19th century, this stub of a road—then known as Laurens Street—was so wretched, residents dubbed it “Rotten Row.”
 
The blight started in the 1830s, when expensive brothels moved into former residences, writes Timothy J. Gilfoyle in City of Eros.