This isn't authoritative but people of the ancient world WERE knowledgeable that these afflictions existed, most notably the origination of syphilis in Europe and when it spread across to the New World.
Actions such as closing down brothels were taken a few years after the big outbreak of syphilis in 1495, but there wasn't an effective cure or therapy for syphilis (Syphilis in Shakespeare's England by J. Fabricus)
(Sidebar 1: I suppose the closing of brothels could be considered a palliative public health measure, but I am not sure if you mean "treat" in the medical sense that only something like medical treatment could provide.
Sidebar 2: There are differing opinions as to the first know origins of the disease and the exact chronology of the spread. Some lay sources claim that common people back to antiquity would inaccurately refer to syphilis as "leprosy"(See STDs--sexually Transmitted Diseases: What You Should Know and how to Protect Yourself). The 1495 spread across Europe is very substantiated, and the other timelines in syphilis existing in the new world before Columbus or it originating with Columbus are debated. )
While I can't really speak to other STDs/STIs, there was no effective treatment at the time. By comparison, one of the landmark syphilis studies in Norway in the 1890-1910 period shows that the disease no longer presented the dramatic effects in terms of being a primary cause of death (The Oslo Study of Untreated Syphilis: An Epidemiologic Investigation of the Natural Course of the Syphilitic Infection Based Upon a Re-study of the Boeck-Bruusgaard Material). This study showed that 90% of the mortality of those who were infected with the disease were not from syphilis as the primary cause.
While syphilis was a contributing factor, it was no longer the more dramatic and traumatic version presented in the book by Fabricus.
This information about the big boom in the European prevalence of syphilis is also agreed upon by John Parascandola in "Sex, Sin, and Science: A History of Syphilis in America", where he spent a lot of the first chapter detailing the first known origin of the disease in Europe.
King Charles VIII the Affable was King of France during the Italian War of 1494, and his troops consisted of a large number of Swiss mercenaries. The confluence of male troops likely aided spread of the disease, but certainly the movement of a large number of people as well as overwhelmingly male nature helped spread the disease across the continent quickly after King Charles VIII the Affable lost.
Quarantines were the norm for suspected lepers, and oftentimes people who contracted syphilis were put into leper homes (p. 556-557 of Twentieth Century Practice: Syphilis and leprosy By Thomas Lathrop Stedman). Even if this was quarantine was inaccurate or semi-mistaken, it still shows a pretty advanced public health measure. Lepers were discriminated against beyond just being quarantined, such as being forced to wear bells ("Formation of a Persecution Society" has an amazing chapter on this).
Sushruta, one of the earliest known text of medical treatise dealing with myriad of diseases(est. 1200 - 600 BCE). He identified 5 different types of Upadamsa which he attributed to sex. He identified those as Vitaja, Pittaja, Kaphaja, Tridoshaja and Raktaja.
Vitaja, Pittaja, Kaphaja are the three main doshas in Ayurvedic tradition[ref: http://online.liebertpub.com/doi/abs/10.1089/10755530152639792]. Dosha are bodily humors. Tridoshas, or three doshas, are diseases that are combination of the three doshas. Finally, Raktaja is conditions relating to bleeding/impure blood.
The symptom of Upadamsa vary from inflammation in the case of Vitaja, to discoloration of genitalia in Pittaja, Itchiness in Vataja and finally bleeding and pus in Raktaja.
The text also identifies promiscuity as one of the reason for Upadamsa in addition to other actions of the individual such as both partners not maintain proper hygiene, masturbation, anal sex, and having sex during periods.
For treatments, the text recommends host of topical treatments by creating a plaster using assortment of herbs and minerals.
In addition to topical treatments, the text also recommends surgical treatments with incisions to draw blood and pus. It even recommends leeches to suck out blood.
Source: Bhishagratna, Kunja Lal, ed. An English translation of the Sushruta Samhita based on original Sanskrit text. Vol. 1. Chapter XII Wilkin's Press, 1907.
The whole book is available on public access.
This text(Sushtura Samhita) and Charaka Samhita form the basis of Ayurveda. Today, Ayurveda is considered to be an alternative for of medicine usually compaed with Chinese traditional medicine. These days, Ayurveda has shunned surgery and has mostly relegated itself to herbal and mineral preparations.
Futher reading: The Bower Manuscript
Can't speak to ancient times, but I can speak to Medieval England, where they most certainly knew about stds. Venereal diseases were common enough to be written about by Chaucer in the General Prologue to his Canterbury Tales when describing the Summoner. There are other records of Medieval writers colloquially referring to stds, most commonly talking about a burning sensation. Most of the time this is talking about chlamydia, as far as we can tell, although the name "chlamydia" was not used to describe the disease back in Medieval England.
It is important to note, however, that although such diseases are referenced, it is not entirely clear that the writers know exactly how the diseases are transmitted. For instance, we can take the example of Chaucer's summoner again, where Chaucer references that the summoner is lecherous. This points out that the disease was caused by his promiscuity, but does not specify which part of the process resulted in the disease. We must be careful in observing that even though Medieval writers may have understood these diseases to be related to sexual contact, the knowledge of how they were transmitted stops there at "something vaguely sexual." There may not have been a clear understanding of how exactly someone could catch chlamydia.
Edit: changed "syphilis" to "chlamydia", which is what I actually meant but for some reason wrote syphilis instead. Also, thanks Mean Girls for my first attempt at spelling "chlamydia" to begin with a "k".
In addition to the other answers, here is a well cited answer about the Ancient Greeks and Romans:
https://www.reddit.com/r/AskHistorians/comments/1t3297/how_did_the_greeks_and_romans_treat_stds/
Yes. In the 1490s, syphilis spread veraciously through Italy, and to a lesser extent, the rest of Europe, killing roughly five million people. Most historians believe it was brought here from America, but it is disputed. From Origins of Syphilis and Management in the Immunocompetent Patient: Facts and Controversies:
The two main current hypotheses on the origins of Treponema pallidum are the “Columbian” and the “Pre-Columbian” hypotheses. Strong evidence supports that Columbus' crew brought T pallidum to Europe at the time of discovery of the New World.
In fact, Pope Alexander VI had a son, Cesare Borgia, who famously had syphilis, which he caught some time in 1497 while in Naples. It supposedly left his face disfigured until 1503, when he got Malaria, which killed his father and nearly killed him; after he had Malaria, his outbreaks in sores came to an end.
Also, syphilis carried many xenophobic terms. In Italy, the H.R.E., and Poland, it became known as the "French Disease", due to the timing of the French invasion of Naples and the outbreak...in Naples. In the Netherlands, it was known as the "Spanish Disease", and in Russia, it was known as the "Polish Disease." The regulation of brothels and general prostitution of this time is due substantially to the spread of syphilis.
To quote Jared Diamond,
When syphilis was first definitely recorded in Europe in 1495, its pustules often covered the body from the head to the knees, caused flesh to fall from people's faces, and led to death within a few months.
People certainly did know they had syphilis as a result of sex, and those with visible postules on their body would be looked down upon. There even existed a famed facial surgeon, Gasparo Tagliacozzi (1545-1599), who specialized in facial reconstruction to rid of any physical evidence of syphilis on one's body.
Until 1908 with the development of the drug Salvarsan, there really were no effective treatments. People would seek Mercury-based cures, which led to the phrase "A night in the arms of Venus leads to a lifetime on Mercury." Herbal remedies were also used, to no significant effect. Treating syphilis was limited to treating the symptoms, and it was far more fatal than it is today. In general, STDs were incurable until long after the Middle Ages had ended.
Edit: More on Cesare Borgia's syphilis, because he was an amazingly interesting man. Cesare Borgia contracted the disease while he was a cardinal, and after he began to have physical symptoms on his face, many reports indicate that he wore a mask in public to cover it up. As to the malarial fever bit, /u/SerLaron answered that curious case quite well. Cesare was lucky in that his doctor would not use mercury to treat him, preferring instead the use of ointments and herbs. Unfortunately for him, his life never really got better after his syphilis cured, but I'm not going to tell that story today.
I have a follow up question. Were any STD strains ever wiped out by Ancient/Medieval rulers? I mean, did any historic rulers ever take drastic measures against people with STDs, such as executing every person they could find who carried a STD, or enacting a strict quarantine on them?
Follow up question if I may. What did people from that time believe caused what we identify today as STDs?
The above answers have addressed syphilis and late medieval times very well. Speaking more towards the ancient world, gonorrhea was known to exist since at least 2500BC in China, and other textbook descriptions resembling gonorrhea existed a thousand years before that.
Prevention was minimal; treatment was normally ineffective (as with any bacterial infection in historical times) and often painful (bloodletting, filling the urethra with warm water, etc.)
The book Deadly Diseases and Epidemics: Gonorrhea gives further details, and is available through Google Books: