I've always heard the story about him having a dream of the cross and being told he'd conquer under it, but I heard recently that by the time he was in power, Christianity had spread to some 50% of the populace in the lower classes, and that by converting to it, he would gain the support of the people against the other elites.
From what we know about demographics and history of that time, is that a reasonable conclusion to draw?
It's difficult to be certain of the motivations of someone, people aren't always fully honest, even with themselves, about their motivations.
Historical religious demographics are also pretty difficult to assess, but the consensus for that time period is much less than 50% Christian, even among the lower classes.
That being said, Constantine was initially acclaimed as Emperor in York (in England) which was in one of the less Christian parts of the Empire, he had personal and family history in the Greek speaking East, which was dramatically more Christian. Perhaps he thought that Christianity would help him appeal to a significant demographic in territories that he didn't control but where he hoped to have support.
His mother was also a Christian, so it's reasonable to believe that she might have had some influence on him.
Remember that later Roman Emperors were more like warlords or dictators, frequently coming into power by forcibly overthrowing the past office-holder and being paranoid the rest of their lives of the same thing happening to them. Taking up the flag of Christianity was as much an astute political move as it was religious-- the Christians were a growing demographic, and removing the sword from hanging over their heads meant that he had at least one source of iron-clad support.
Bear in mind that he didn't actually convert until nearly the time of his death. He openly supported the religion, favored it and removed laws that persecuted it, but he didn't technically become a Christian until the last minute.
Theories vary widely as to why exactly this was the case. My personal belief is that he saw no reason to abandon the old Roman tradition of syncretism, and likely viewed the Christian God as some variety of Sol Invictus or other supreme deity; he was quite happy to portray himself publicly as a divine Emperor, in line with long Roman tradition.
However that's a major no-no for Christians. So his late baptism was probably (again, IMO) a matter of either being gradually worn down by believers in his various social and political circles over time, and a gesture of final commitment to what had become one of the more important demographics supporting his reign.