To answer your question, I'll make a few points clear: the fall of the Roman Empire happened traditionally in the year [476] (http://www.ushistory.org/civ/6f.asp). Yet, we can also interpret this date in a more wide time frame, since Rome had been showing signs of weakness since the late 4th century. At this point in time, the official and main religion was Christianity, legalized in 313 by Constantine with the [Edict of Milan] (https://en.wikipedia.org/wiki/Edict_of_Milan) and made official state religion in 380 Emperor Theodosius with the [Edict of Thessalonicaby] (https://en.wikipedia.org/wiki/Edict_of_Thessalonica). The Roman Pagan religion was outlawed shortly after and declined rapidly. Emperor Marcian decreed, in the year 451, that those who continued to perform the pagan rites would suffer the confiscation of their property and be condemned to death. Although paganism survived in modest numbers for a few centuries afterwards, it is safe to say that by 450 the vast majority of inhabitants of the Roman Empire were Christian.
At this point, we can come to your question. People were definitely religious, especially for modern standards. Christianity had gained much momentum, especially with the approval of the Emperors. The Christianization of the Roman Empire became complete when the emperor Anastasius, who came to the throne in 491, was forced to sign a written declaration of orthodoxy before his coronation (Note that in this case we are talking about the Eatern part of the Roman Empire, that survived until 1453).
The Western part of the Empire, after 476 AD, was invaded by nomadic (also called barbarian) populations. Interestingly, this moment was a fundamental moment of Christianity. This is when the papacy evolved and took form, and the Church was not yet divided between orthodox and Cahtolic. Also, this is when monasticism was founded in the West by [Saint Benedict] (http://www.ewtn.com/library/MARY/BENEDICT.htm). The barbaric populations that invaded Europe converted to Christianity and boosted the religiosity of the people. Christianity spread to Northern Europe (especially in Britain and Ireland). A great deal of religious architecture was accomplished in the [pre-Romanesque style] (https://en.wikipedia.org/wiki/Pre-Romanesque_art_and_architecture). This is the peak of the patristic literature, with great theologians like St. Auugustin, St Gregory, St. Ambrose etc...
In summary, during the fall of the Roman Empire (a period more properly known as Late Antiquity), people were very religious and the momentum and influence of Christianity was at it peak. Fundamental institutions like the papacy, monasticism, ecumenical councils, and patristic theology.
Sources :
[The Power of Religion in Late Antiquity] (http://www.ashgate.com/default.aspx?page=637&calcTitle=1&isbn=9780754667254&lang=cy-GB)
[Christianity, Empire, and the Making of Religion in Late Antiquity] (http://www.upenn.edu/pennpress/book/14462.html)
Thanks /u/jamgoth for specifying the time period (5th century). The religious landscape of the 5th century Roman Empire was very different from that of the 3rd and 4th. By the 5th century, Christianity was fully accepted in the Empire. In contrast to the persecution of Diocletian in the late 3rd century, Christianity now enjoyed favored and mostly exclusive control of the religious life of both East and West. Several emperors did try to revive paganism, Julian being the prominent example; nevertheless, his untimely death in 363 ended the hopes of widespread pagan restoration across East and West. Even in his time, his pagan views were widely mocked. Good examples are his pagan speeches in public spaces, such as Antioch, where the locals often derided his intellectual paganism. Another example is the philosopher Hypatia, who was murdered by an anti-pagan mob in Alexandria in the 5th century.
Although the (Neo)platonic academy of Athens would not close until Justinian’s decrees in the 6th century, its influence as a pagan institution had waned dramatically. Interestingly, aspects of neoplatonism were incorporated into the orthodox* interpretation of Christianity, such as the immortal soul and the logos. Augustine, a notable Christian theologian, reinterpreted Plato and Aristotle to fit into the Christian philosophy.
Meanwhile in the West, the Germanic tribes entering the Empire were Arian Christians, who followed the teachings of Arius — a heretic in the eyes of the orthodox church since the First Council of Nicaea, as convened by Constantine I. This did cause conflict with the emperors in Cosntantinople, since heretics posed a threat to the legitimacy of the Church, which at this time was heavily tied to the Roman state, legitimizing the authority of the Emperor as the representative of Christ on earth.
As for the fervor of the people at large, we have to remember that during this time period, particularly in the West, the opportunities for young men to advance in the imperial bureaucracy or military had declined in parallel with the weakening of the Western state. Increasingly, these individuals turned to monastic orders. We see the rise of monasticism starting in these centuries as a result. Furthermore, emperors often used local bishops as de facto administrators for cities.
Lastly, and most importantly, the large number of heretical movements and resulting church councils illustrates a rise in the fervor of Romans, both East and West. Arians, Nestorians, Monophysites, and other groups that disagreed with the orthodox authorities found themselves persecuted by the imperial administration, not to mention outright pagans. The Christian mobs and the accompanying imperial persecutions show a population and an empire that took religion very seriously. So to conclude, the uptick in official and unofficial persecutions as well as religious councils stem from a highly religious population, who increasingly turned to Christianity for spiritual answers to the violent and chaotic world they lived in.
Dr. Stephen Marrone — The Early Middle Ages — Tufts University.
Dr. David Proctor — The Byzantines and Their World — Tufts University.
Deno John Geanokoplos — Byzantium: Church, Society, and Civilization Seen through Contemporary Eyes.
The First Council of Nicaea - 325 AD.
Could you expand upon "fall of the Roman Empire?" Do you mean the "end" of the Western Roman Empire in 476, or the capture of Constantinople in 1453?