Of American-English origin, the phrase red-light district denotes the part of a town or city in which prostitution and other commercial sexual activities are concentrated.

This phrase refers to the use of a red light as a sign outside a brothel.

Here, red light denotes a warning, a signal to desist in a course of action or thought—as is clear from the following from On the Protection of the Innocent and Helpless Members of the Community from Venereal Diseases and their Consequences, by Albert L. Gihon, Medical Director U.S. Navy, published in Public Health Reports and Papers. Volume V. Presented at the Meetings of the American Public Health Association in the Year 1879 (Boston: Houghton, Mifflin, and Company, 1880):

What do we, the foremost race in Christendom, do to repress prostitution? Nothing. Virtuous nothing? No,—criminal nothing. The higher we rank in self-complacent godliness, the less we do. We see the streets, especially the fashionable thoroughfares, thronged with harlots, conspicuous by their natural beauty or their skillfully added charms, by the extravagant richness and fashion of their attire, by their magnificent equipages and their immodest demeanor. We meet them in the theatres, the concert saloon, the ball-room, and at the hotel-table. We elbow them in the great shops and stores, and we pull our wives and daughters aside, in the hope that their feminine curiosity will not induce them to inquire, Who are these gay birds, of such brilliant plumage? If they do not learn from each other, if the mother neglects to instruct her daughter, the morning papers will give the street and number of Madame So-and-So’s splendid establishment, which was raided by the police the night before, and its elegantly attired inmates brought before a magistrate, fined, admonished, and discharged to return to their former abode, or, if that has been broken up, to be scattered to a dozen new sites.
Is it not better that the red light and conspicuous number should mark the door of the strange woman, “in the twilight, in the evening, in the black and dark night;” that the closed blinds and sombre stillness should silently say, “My son, remove thy way far from her; come not nigh unto the door of her house, lest thou give thine honor unto others and thy years unto the cruel,” than that the brazen affectation of respectable callings in reputable neighborhoods should make it possible for men and women to frequent them without concealment and without fear of detection?

It seems that, in early use, red light was a proper name for a brothel—the earliest occurrences that I have found are:

1-: From The Detroit Free Press (Detroit, Michigan) of Friday 26^th March 1869:

Defaulting Witness Captured.—Flora Fleming, a former inmate of the “Red Light” establishment, who, rather than appear as commanded in the Recorder’s Court to testify in the case of Norman Everson against the People, in an appeal from a judgment rendered in the Police Court, went across the border, was captured yesterday and lodged in jail to await the sitting of the court.

2-: From The Kansas City Times (Kansas City, Missouri) of Friday 4^th May 1877:

Under the head of a “Wail From Out the Darkness,” the Ft. Worth Standard * gives one of the most touching stories from real life that we have read in many a day. Here it is: “Nellie Collins, an inmate of the ‘Red Light’ dance house, is lying at the door of death. Under the effects of a ravaging disease and general exposure, her delicate constitution has given way, and she is a wreck of what was once a beautiful and intelligent lady. Realizing her situation, she sent for a clergyman, and besought him to supplicate the throne of grace for the pardon of her sins and restoration of her health. The clergyman told her that the Savior, when in the world, pardoned prostitutes, and could pardon her. She asked him in the event of her death to convey her remains to the church and preach a sermon warning others against her shameful example. She implored that when death was upon her, she might be carried forth into the street, so that she might die outside of a house of bad reputation. Truly a sad case. How many hundreds and thousands there are in those haunts of sin who are unwilling to die there, and would leave to-morrow if they had the right kind of sympathy?”

* I have not found the original article in the Daily Fort Worth Standard. Curiously, one Nellie Collins appears in the next quotation:

3-: From the Daily Fort Worth Standard (Fort Worth, Texas) of Saturday 21^st July 1877:

TlRED OF LIFE.
A Fair but Frail Creature Attempts Suicide.

Minnie Levi, a demi-monde, at present an inmate of the house known as the Red Light, attempted to commit suicide last night, by taking morphine. About 10 o’clock she asked a hack driver to come up town and buy twenty-five cents’ worth of morphine for her. After he returned she, in company with another woman, named Nellie Collins, got into his hack and were driven around town. Before returning to the house they went to several places in search of some one—probably a male friend—but did not succeed in finding him. Soon after returning to the Red Light, Minnie requested the hack driver to purchase more morphine for her, stating that Nellie had taken what she had away from her, and that she was suffering so much from a pain in her side that she could not do without it. The hack driver then went to a drug store the second time and purchased two bits worth of morphine. About 12 o’clock the other inmates discovered that she had poisoned herself, and immediately sent for Drs. Burts & Field. She was then in a very precarious condition, on the very brink of the dark valley, as it were, and but for the timely arrival of Dr. Burts, would certainly have crossed over. The necessary remedies were administered, however, and she was soon placed out of danger. The deed was no doubt premeditated, the unfortunate creature being impelled to commit the rash act by blasted hopes, disappointments, and the concomitant miseries of a life of shame and degradation. At last accounts, she was doing well, though suffering considerably, and will recover speedily. The world would probably have been benefitted by the accomplishment of her purpose, as the future is barren of even the shadow of a hope of her reformation, unless she experienced a sufficient foretaste of death to cause her to repent.

4 & 5-: From City News, published in The Leavenworth Times (Leavenworth, Kansas):

4-: Of Saturday 24^th May 1879:

The proprietors of the “Red Light” bagnio were arrested yesterday by officer Yerkes, charged with keeping a disorderly house.

5-: Of Sunday 25^th May 1879:

The trial of George Walker and Mariah Bland, of the Red-Light bagnio has not yet taken place, on account of the arrest of the former at the instance of Atchison authorities for burglary.

6-: From the Brenham Weekly Banner (Brenham, Texas) of Friday 30^th January 1880:

At Sherman on Wednesday, city marshal Ball attempted to eject a rough gang from the Red Light, a brothel. One of the gang Alf. Johnson, when outside the door fired at Ball wounding him slightly. Ball then fired shooting Johnson through the heart, at the same time one of Johnson’s friends fired at Ball shooting him through the lungs and inflicting a mortal wound.

The texts containing the earliest occurrences of red-light district that I have found indicate that the phrase was originally used of Louisville, Kentucky:

1-: From the story of a murderer called Albert Wing, published in The Cincinnati Enquirer (Cincinnati, Ohio) of Monday 21^st August 1893:

Wing took his young wife to the bordello of Madam Mertie Edwards, on West Green street, the red-light district of Louisville.
                    THERE HE LIVED
Off her shame and loafed. She was still rich in beauty, a fatal dower, and used to delight the callers with flowery praise and voluptuous song. She was an expert with several stringed instruments and the piano besides, having a cultured voice.

2-: From The Cincinnati Enquirer (Cincinnati, Ohio) of Friday 12^th January 1894:

TRIED TO CREMATE HERSELF.

Louisville, January 11.—Mary Payne, a ten-year-old girl, attempted to burn herself to death at the Colored Orphans’ Home because told she was going to be returned to her home. Several days ago she came to the Home and her body was covered with frightful scars, cuts and bruises. She told a harrowing story of brutal treatment by the woman who claims to be her mother. The woman demanded possession of the child, and the Home authorities consented and so notified her.
The result was she went into the kitchen and deliberately allowed her clothing to ignite. The flames were extinguished, but not until she had been dangerously burned. The child declared she would kill herself in some way if forced to return home, and the Home authorities have determined to keep her even if there is a fight in the courts. The child’s home is at 611 Tenth street, in the heart of the red light district.

3-: From an article about the Rescue Mission of 715 West Green street, published in The Courier-Journal (Louisville, Kentucky) of Thursday 19^th July 1894:

Some four or five months ago curiosity attracted the attention of a Green-street courtesan to the woman who dared to give up her home for the “red-light” district, with little else than prostitutes for neighbors and nothing to see from her front windows but sin and vice. At first a passing glance and a jesting remark were all that were bestowed upon the little sign, “Services for Women Every Evening at 7:30,” but from day to day the scoffing diminished and a real interest began to manifest itself. The woman soon saw happy faces in the little mission, taking the places of sin-hardened features, and she began to wonder if such a change could ever come over her own disfigured countenance. She attended some of the services and soon threw her whole soul into an effort to reform, and in a short time she became converted, and to-day she is a happy woman and a respected citizen.