It is not a modern phenomenon; rectal prolapses can occur in infants who have problems with constipation or diarrhea. Medical history is not my specialty so I don't know the full history of the condition. But because of the aforementioned reason, it was included in Nils Rosén von Rosenstein's 1764 textbook Underrättelser om barns sjukdomar och deras botemedel (English title: The diseases of children, and their remedies ), which was a popular book, translated into 10 languages in the 18th century, which is considered history's first textbook of pediatrics. It doesn't seem to be digitized yet but I have a photo of the relevant page. It reads in full (my translation):

§3 On the falling out of the rectum.

When children have diarrhea, it often happens that the rectum goes out, which then becomes a habit if not cured immediately. I have not found a better cure against this, than to dab it with a fine sponge, which has been dipped in warm wine. Fine soot, or crushed and sifted bark usually helps as well, when strewn upon the intestine which is then carefully [re-]inserted. Likewise it is good to smoke the intestine with [mastic](https://en.wikipedia.org/wiki/Mastic_(plant_resin)).

Rosén does not explain his rationale for these treatments, but they seem fairly straightforward in terms of the Humorism thinking of the time, where the body was governed by the balance of four 'humors' or bodily fluids. In that model, diarrhea means an excess of moisture, or phlegm. I.e. the patient is phlegmatic (a term that lives on not for the diarrhea but the mental states they associated with the condition). The 'cure' for an excess of moisture is then to administer things that are dry, such as the dry powders and smoke.

On a related note, the administering of tobacco-smoke enemas with bellows to drowning victims would become a common practice in about the same period. Since drowning is, when viewed through the lens of humorism, another problem stemming from an obvious excess of moisture.

Short answer:

There's a kingly gloire to the history of proctology, and while we only have about two centuries of historical documentation for the treatment of rectal prolapse, there's good reason to think that treatment has been attempted for much longer as we do know that disorders of the rectum generally have been treated since antiquity.

Discussion:

No discussion of historical proctology can begin without mention of Louis XIV's fistula, treated by a surgery without anesthesia which was the talk of society; Lully conducted a Te Deum afterwards, and his physicians leave us a surprisingly detailed account in the Journal de Santé du Roi. A fistula isn't a prolapse, still, we know that physicians were devoting a great deal of attention to piles and other complaints of the area-- no reason to think that they didn't see a prolapse on occasion, the accounts we have are often anatomically vague and we can't always tell just which condition is being described.

The first description I can find for a specific treatment for prolapse is 1831; work of the pioneering surgeon Frederick Salmon. He discusses prolapse as distinct from intussusception, which appears to have been known to and treated by practitioners from antiquity (likely without success). Salmon founds “The Infirmary for the Relief of the Poor Afflicted with Fistula and Other Disease of the Rectum,” and he claims a surgical cure for prolapse, this seems to be the first recorded discussion of this as distinct from other complaints in the same area. Salmon makes clear that earlier physicians had seen and attempted to treat it, but had not properly understood the anatomy.

Gregory Tsoucalas, a contemporary (to us) physician and medical historian wrote recently of Salmon in Surgical Innovation

Although [Salmon's] work was too significant for the era, he was almost completely neglected by historians, most probably due to his clash with his fellow surgeons, who had been considered by him as scientifically inadequate in anorectal diseases.Frederick Salmon succeeded to present a pioneering restoration of the rectal prolapse, announcing zero fatality. Despite his persistence toward the hepatic “cure,” his observations about the causes and symptoms show great similarity with the modern views. Salmon is now considered as one of the most important figures in proctology.

Prolapsed rectums are not exclusively "modern phenomena", indeed they're not exclusively human phenomena, veterinarians see them too.

While we don't have a discussion of Hippocrates or other classical physicians treating rectal prolapse, we do know that they did proctology. We have Hippocrates' "On Fistulae" as evidence of their familiarity with and treatment of these conditions, and we have specula and dilators from classical antiquity which were used to treat complains of the anus and rectum. So although we don't have a specific description of their use in prolapse, we do know that physicians of classical times were treating other conditions affecting the same parts.

The celebrated 14th century English surgeon John of Arderne treated fistula as well:

I cured a man from Northampton of the aforesaid Fistula in ano who bad three openings in the left buttock and three in the cleft of the scrotum as is here depicted, and all in turn communicated the one with the other and had perforated through the middle of the scrotum. I cured him by incision of all the holes, both in the rectum and elsewhere.

None of these early physicians and surgeons give us a clear description of a prolapse; the most we can say is we know that they must have happened, we know that patients are symptomatic and would be likely to seek care, and that there were physicians who addressed proctological complaints generally.

A search of sources in Chinese and Ayurvedic medicine doesn't turn up much-- the Wu Shih Erh Ping Fang ("Prescriptions for Fifty-two illnesses) of 168 BCE has a discussion of hemorrhoids, though we can't be sure that nomenclature doesn't include prolapse and other disorders. In the Islamic world, al Tabari, the great scholar of the 9th century CE has a compendium of medical literature ("The Paradise of Wisdom") which includes a section on diseases of the anus and fistula in particular.

So: Before 1820, we can't find a clear description of a treatment for rectal prolapse-- if you were unlucky, it's possible that someone might have attempted surgery, which would be unlikely to have happy results. Fistulae and hemorrhoids are problems more readily addressable with the means that early physicians had at hand.

Sources:
Wu, J. S. (2009). Rectal Prolapse: A Historical Perspective. Current Problems in Surgery, 46(8), 602–716. doi:10.1067/j.cpsurg.2009.03.006 

An Operation For Fistula In Ano Two Hundred Years Ago. The British Medical Journal, Vol. 2, No. 247 (Sep. 23, 1865), pp. 310-311 [there are more modern sources for the story of the celebrated operation on Louis XIV, but this is colorful and correct as to details)

Salmon F. Practical Observations on Prolapsus of the Rectum. London: Whittaker,Treacher and Arnot, Ave-Maria Lane; 1831; chapter I, p B.

The History Of Rectal Surgery. The British Medical Journal, Vol. 2, No. 3699 (Nov. 28, 1931), p. 999

Tsoucalas, G. (2018). British Surgeon Frederick Salmon (1796-1868) and His “Trans-Fixing Pins and Excision” Surgical Procedure for the “Rectum Prolapsus.” Surgical Innovation, 25(1), 88–89. https://doi.org/10.1177/1553350617731384

Bliquez, Lawrence J. “Greek and Roman Medicine.” Archaeology, vol. 34, no. 2, 1981, pp. 10–17. JSTOR, www.jstor.org/stable/41727119.

Bliquez, Lawrence J, "The Tools of Asclepius: Surgical Instruments in Greek and Roman Times" [2014]