The theory that Christianity caused the downfall of the Roman Empire is largely an Enlightenment-era idea, one that would have been advanced by thinkers such as Voltaire and Denis Diderot. However, the most well-known advocate for the notion comes from the English historian Edward Gibbon, who wrote the monumental The History of the Decline and Fall of the Roman Empire, originally published in 1776. His theory was that pagan Rome had the ability to fight because their religion supported it--Mars, for example, being one of the chief gods and the divine forebear of the Roman Empire, was also one of the various gods of war, along with figures such as Quirinius and Bellona. Under the guidance of pagan figures such as Augustus, Trajan, Hadrian, Marcus Aurelius, etc., the Empire had expanded to control an enormous amount of territory, and their martial ability could be seen as either coming from the gods or being done in homage to them. Once Christianity became a protected religion under Constantine in 313 and ultimately the official religion of the Empire under Theodosius II, Gibbon saw the Empire retreating in its territory and becoming more "complacent," possibly because Christians were not concerned with building a glorious state in this life since they'd be going to one in the next. Additionally, Gibbon claimed that Christians could not serve in the military, so the legions began to lose troops, thus reducing the Roman military machine, which ultimately led to the downfall of the Empire.

In terms of the validity of the question, it's obviously up for serious debate. In terms of fact, Gibbon was absolutely incorrect about soldiers--while some of the great Christian thinkers of the early Church said that Christians could not have jobs in Roman society that would support the pagan Roman state, once the religion became more accepted, we know that there were many Christians that served in the military. Furthermore, Gibbon took a very western-empire-focused look at his history, saying that the fall of the Western Empire in 476 was the end of Roman civilization; he ignored the Byzantine Empire, which was profoundly Christian, to the point that a few of their emperors were essentially monks in their palaces, and lasted until 1453. There are also claims that could be made about the Christian character of the Holy Roman Empire and other similar western states, but ultimately Gibbon's views are problematic at best, and so his book itself is regarded as a "classic" by scholars more for his methods and encyclopedic understanding of the Empire, rather than for any actual claims he made.