First, I recommend that you or anyone else interested in the context of what Gnosticism is and where it came from read my response and the followup discussion to this question on Gnosticism and other early sects. That answer also links to an earlier answer to this question, where I give an overview of the major divisions of Christianity that you may find enlightening.

As for your current question on the end, rather than the origins, of Gnosticism, I'm afraid I'm going to have to disappoint you by resorting to the two most common but unsatisfactory clichés used by historians: we don't really know, and it kind of depends. "We don't really know" is really the most accurate here. There are no surviving documents that detail exactly what happened to any of the various extinct Gnostic groups. There is no diary entry or letter that begins with, "I am the last surviving Sethian, and here's what happened to the rest of my congregation…". But there were many different Gnostic groups, so the answer is probably also "It kind of depends." Some groups probably slowly withered away, while others likely faced persecution. Complicating matters further, the Mandaens, who may represent one of the earliest Gnostic groups, still exist today. So arguably the Gnostics haven't died out at all.

It seems that for most groups, in most places, most of the time, the route to extinction was more gradual and organic, and less sudden and violent. The ancient world was a place with a great diversity of religions. It was not at all unusual for them to attract followers and grow over the decades and centuries, only to see their members become attracted to other religions. The height of Gnosticism seems to have been in the second and third centuries CE. By the end of the third century, the popularity of most Gnostic groups seems to have been fading, at least based on the amount of writing Christians were doing about them. It probably should not be taken as mere coincidence that the decline of Gnosticism seems to have occurred at the same time that Christianity was the fastest growing religion in the Mediterranean. So, part of the reason Gnosticism died out was that people left it and became Christians. Augustine of Hippo, one of the most influential Christians in history, was a Manichaean in his youth (Manichaeism, depending on your preferred definition, either is a Gnostic religion or is dualistic with a strong Gnostic influence). But also bear in mind that people don't have to actively leave a religious group for that group to disappear. Simply recruiting fewer and fewer converts over each successive generation will eventually cause the group's extinction as the active members grow old and die.

There likely was some persecution involved, though. Gnostics were caught up in the same persecutions that affected Christians. A local official who found a member of a minority religion wasn't likely to dig too deeply into their specific theology before insisting that they make sacrifices to an image of the emperor. As for why Christianity continued to thrive, and even increased in numbers, even during the most intense phase of persecution in the early fourth century, while most Gnostic sects did not, well, that's an open question.

One thing I will say about violent persecution of Gnostics is that, if it did occur in an organized way, it probably was not the Christian Church that did it. "The Church," as a powerful, centralized organization capable of calling a crusade against a rival religion, did not exist until the Middle Ages. That's hundreds of years after Gnosticism had vanished from Europe. Not to excuse the Church for the violent persecutions and crusades that it did call - we just can't lay blame for this one.

You mentioned Constantine and his embrace of Christianity. Much has been said over the last 1700 years of the influence Constantine had on Christianity, and the influence Christianity had on Constantine. But without delving too far into it, I think it's fair to say that Constantine did not choose Christianity on a whim. He didn't just pick randomly from a grab bag of religions and come up with Christianity. Christianity was rising in the early fourth century. Gnosticism, as I said earlier, was declining by then. Now, when I say "declining," I don't necessarily mean it was nearly extinct. There is evidence of Gnostic groups thriving into the sixth century, and later in some cases (once again, Mandeanism is still an active religion today). If we consider Manichaeism to be Gnostic, then it's worth noting that that religion wasn't even founded until the middle of the third century, and didn't reach its peak until perhaps the fifth. Nonetheless, at the time of Constantine, Christianity was ascendant in a way no other religion was.

We try not to speculate too much around here - "If Constantine had embraced Gnosticism, would the Catholic Church today be Gnostic?" isn't a question I can answer. Constantine didn't embrace Gnosticism, and the Catholic Church isn't Gnostic. But even wondering it might give Constantine more credit than he really deserves. Christianity was gaining popularity before Constantine, and Constantine's moves towards Christianity (he did not make Christianity the state religion of Rome - that wasn't until decades after his death) did not magically make the Empire uniformly Christian. It was another century or more before Christianity was undisputed as the dominant religion in Europe and the Mediterranean (longer if we are talking specifically about Nicene Christianity). The long term "success" of Christianity and "failure" of Gnosticism, if I can use such loaded terms, is due to far more factors than the religious policy of a single emperor.