The basic story comes from the diary of Edmond and Jules Goncourt. The final edition includes the following for the date of 2 May 1885 (translation mine; beware of the rude language):
It seems that the night before Hugo's funeral, a night of sorrowful wake for the people, was celebrated by an enormous copulation, by a priapée [orgy] of all the brothel women on holiday, who fucked with anyone on the lawns of the Champs-Elysées, like republican weddings that the good police respected.
Another detail about the big man's fuck-funeral [funérailles foutatoires in French] and the detail comes from the police. For eight days now, all the Fantines of the gros numéros [Fantine is a prostitute in Hugo's Les Misérables and the "big numbers" are brothels, then identified by a large number above the door] have been working with their natural parts wrapped in black crepe, their cunt in mourning. (source)
As we can see, the Goncourts report hearsay (Il paraît...) and a police report that may have been second- or third-hand. They were also annoyed at the idolatry towards Hugo, so they may have exaggerated a little bit.
The first edition of the Journal des Goncourt, published in 1895, only includes the second part in shorter form (here).
Now, what actually happened?
The daily La France of 3 June 1885, reporting on the funeral, writes that, during the night,
the mud of vagrants, the scum of racetracks and night clubs, the gamblers, the bookmakers, the whores, arrived. This mob of drunks, shouting, singing, laughing, caused a scandal.
The journal claims that they tried to go under the Arc de Triomphe but were repelled by the police. The crowd then booed Hugo and the police (source).
Another source is the catholic daily La Croix, on 3 June. Note: La Croix hated Hugo. Under the title "Shameful bacchanals", it describes an unruly crowd of street peddlers, wine merchants, drunks etc., and cites another paper that notes the "lack of contemplation". Then:
Some gangs even try to organize merry farandoles and while drunk people lie down on the lawns, groups indulge behind the bushes bordering the new avenue Victor Hugo in abominable outrages that the police are powerless to repress. (source)
There were lots of prostitutes and brothels in late 19th century Paris (see Gonzalez-Quijano's PhD, Paris Capitale de l'Amour, 2015). What seems plausible is that brothels and independent prostitutes, just like other professions, tried to make as much money as possible from the 2 million people who participated in this unique event, and went to work where their customers were. Famous brothels like the Chabanais did publicity stunts, such as sending their girls distribute flyers in front of the Opera (source), so we cannot rule out that some did the black crepe thing. And drunk people certainly had sex in public during that night.
The "brothels closed down because Hugo liked prostitutes" story, however, seems to be a nice tale derived from the facts above and inspired by Hugo's legendary sexual appetite.
I have an earlier answer, if you're interested!
Victor Hugo might have been the consummate womanizer. While married (to Adele Foucher), he cheated on his mistress (Julienne Gauvain, stage name Juliette Drouet) with another mistress (Leonie Biard). For seven years straight. And tried to cheat on Biard, who was married, with another actress named Alice Ozy.
Who, er, at the time was sleeping with Hugo's son. Right then.
So it will come as no big shocker, I'm sure, that Hugo's involvement with the ladies did not stop at his multiyear or even adulthood-long, deeply forged and felt relationships. However, the story about the Parisian brothels closed for Hugo's funeral is probably a myth.
The usual source cited is the journal of Edmond de Goncourt, a contemporary author and no big fan of Hugo. Goncourt has a few things to say about what he considers the ridiculous and undeserved pomp of Hugo's funeral. First, that the adulation is all political and not actually related to the literary merits of Hugo's work. Second, and hilariously, he derides the public enthusiasm for Hugo as "l'hugolatrie"--Hugo-idolatry, parodying the recent coinage "Bardolatry" to describe quasi-worship of Shakespeare.
And Goncourt wasn't wrong about at least the extent of the festivities--one to two million people are said to have lined the streets of the public procession. Compare that to the population of Paris, which in 1885 was probably around 2.3 million. Accounting for contemporary exaggeration, the point is, that's a lot of people.
Goncourt does give one additional detail, and it does pertain to prostitutes. However, they're not mourning in the, well, classical sense. According to Goncourt:
Concerning these funeral festivities--a curious detail given by the police--in these nights of priapees [rampant sexual activities, presumably], all the prostitutes worked on the lawns of the Champs-Elysees [famous Paris street], their private parts surrounded by scarves of black crepe-cloth.
This is the rumor that Goncourt chooses to report: prostitutes mourning, but not quite like that. If the Paris brothels were "shuttered," well, they were shuttered because their sex workers--along with legendarily a good chunk of the Paris population--were taking their business outside. I suppose it is up to you how seriously to take Goncourt and his claimed source.