I can answer this for you.
First of all, a little clarification--I can answer this as it pertains to Parisian prostitution in the 19th C. My expertise is on prostitution in Britain, but because Britain had no registry system, details of this kind are hard to come by, and hard to verify.
However, it wasn't like that everywhere. In fact, France had a registry system for prostitution, and even better, they had a guy called A. J. B. Parent-Duchâtelet--he was a tireless researcher in Paris who wrote a groundbreaking study of prostitution in the 1837. Fyi, I wrote more about him here.
Anyway, in his study on Parisian prostitution, he has this to say RE: babies & brothels.
When a prostitute becomes pregnant, all her comrades are kind and attentive, particularly during and after confinement, when their tokens of affection and interest are redoubled: they dispute with each other as to who shall wash the child's clothes, and who shall take care of the mother. The child does not want for nurses: all wish to have it, and the mother is no longer its mistress. In fact, this class of females are much more anxious to nurse their children than unmarried girls who become mothers, but are not prostitutes, and even than many others who move in respectable society. ....They despise and renounce those who take no care of their children, and attach a certain degree of glory and pride to the accomplishment of their maternal duties. pg. 70
He goes on to say that many of these children die--which was not all that unusual for all children at this time. However, "as to their education, some mothers are careful of their actions in presence of their children, bring them up well, and do all they can to rescue them from their own condition, and even keep them ignorant of it. I am happy to say that most of those who bring up their children, act in this manner."
Parent-Duchâtelet also noted that "prostitutes are much more apt to conceive than is generally admitted: in order to this it requres the concurrence of the will and the laisser-aller [lack of restraint] of the girl, an intellectual and moral state, foreign to the habitual exercise of her vocation. And if these girls then miscarry, it is either from criminal measures [abortion], or from the exercise of their vocation.
Some prostitutes are singularly prolific, having 7, 8, and even 10 children: but when this occurs, they are generally married or attached to one man; in these cases, their labors are fortunate, and their children are as lively as other children." (pg. 94)
Mortality, though, was dangerously high. He also notes in that section that of ten children born to prostitutes in hospitals, half died at birth, the other half during their first year. Though many prostitutes did have multiple pregnancies, their chances of bringing a child to adulthood were slim, unless they were what you might term courtesans or high-class prostitutes.
None were ever able to replicate Parent-Duchâtelet's study on prostitution, though it generated incredible interest and many attempts to understand prostitution elsewhere, especially in Britain.
Alain Corbin's Women for Hire: Prostitution and Sexuality in France after 1850, (published 1996), is another great secondary source, and a fantastic history, as well.
Edit, words and such.
This is a popular question around here, feel free to check out these earlier threads with answers:
There has been an excellent archaeological study done on a known brothel from New York's Five Points district. The brothel was in use in the early 19th century. Of special interest was during the excavation of the brothel's privy where remains of a 20-22 week old fetus were discovered mixed in with the trash thrown into the privy. The implication here being either an abortion or a miscarriage, but certainly an event that someone attempted to hide by disposing the fetus in the privy.
This is not the only instance of fetal or infant remains being disposed in brothel privies. There have been remains discovered in privies associated with late 18th and 19th century brothels in both Minneapolis and Philadelphia, but in both cases the context of the remains was disturbed. What that means is that in these two other cases we can't be sure the infant and fetal remains were intentionally disposed of in the privies, or if they were accidentally added to the privies later on, perhaps from an unintentionally disturbed burial used to fill in the privy. The case from New York, on the other hand, is very clearly an intentional disposal of the remains in the privy.
Regardless, this is a very small sample size and doesn't really tell us much about patterns of behavior towards abortions, miscarriages, and infanticide in 19th century U.S. brothels. What it does tell us is that, at least in one case and quite likely others, there was some sort of social stigma against abortions or miscarriages, or a stigma against pregnancy in prostitutes that would make it difficult for them to gain customers, that resulted in at least one case (and possibly more) where an abortion or miscarriage was intentionally hidden.
It may be that the situation was on the whole much more like what /u/prehensilefoot is describing for Parisian brothels, but this discovery does at least raise the possibility that there might have been some conflicting attitudes about children and pregnancy among prostitutes.
Edit: I should clarify that the context of these brothels, unlike the Parisian brothels, is among the complete illegality of prostitution. Indeed, the brothel shut down after police action was taken in the 1830s. That context might be important for understanding the decision-making that went into hiding an abortion or miscarriage in a privy. Additionally, the availability of medical care (or lack thereof) alongside high rate of mortality among mothers, not just infants, may also significantly factor into decisions made about abortions.
Crist, Thomas A. 2005 Babies in the Privy: Prostitution, Infanticide, and Abortion in New York City's Five Points District. Historical Archaeology, 39(1):19-46.