It comes down to whether you subscribe to the 'decline' school or 'transformation' school of Late Antiquity historians.

The 'decline' school is led by A. H. M. Jones, with his work The Later Roman Empire, 284-602 (1964) being a continuation of Gibbon's view that the Roman Empire was in 'decline', although in a much more neutral and balanced approach. In his work, he follows Gibbon's critique of Christianity and identifies three main adverse effects of Christianity - economic drain on the state’s resources, source of social tension/disaffection and moral/morale decay (Jones 1964: 1046-7, 1062-4).

In the other corner is P.R.L. Brown, with his work The world of late antiquity, AD 150-750 (1971), which denies that the fall of Rome was significant at all - namely, Rome did not 'decline', but 'transformed'. Brown and the 'transformation' school have a much more positive view of Christianity during the period.

For the financial drain of the state’s resources, Jones identifies Clergy and monks as ‘idle mouths’, with large endowments of money and land given to churches, and lots of well-paid clergy who are 'economically unproductive'. This has been countered by the 'transformation' school that Church and clergy still pay land taxes, and that Jones’ claim of the church costing more than administration is difficult to substantiate, as there is very little data for church revenues before the sixth century. Peter Brown has much to say about this topic in another work of his, Through the Eye of a Needle: Wealth, the Fall of Rome, and the Making of Christianity in the West (2012) (which is a great read by the way).

For Christianity being a source of social tension/disaffection, the 'decline' school argues that Emperors try to enforce orthodoxy, which often leads to criminalisation of alternative views as ‘heresy’ or ‘paganism’. However, the counter argument is that laws not rigorously enforced, which you can see even in the so called 'Great Persecution' of Diocletian. There was also little sign of religious motivations in civil strife, which ironically played a larger part in the more stable East than the declining West.

Lastly, we come to the most controversial issue: moral/morale decay. This is a highly subjective topic, with the decline school putting out two main points - a decline in public spending/spirit, and a lack of 'patriotism' and resistance against the 'barbarian invaders'. The main problem confronting historians is can we attribute these declines to Christianity at all, and whether there was actually a decline in both public spending and resistance. One argument against the lack of resistance is that there was little tradition of organising local defense - with the collapse of the Rhine frontier defense, 'professional' Roman forces had melted away, and we mostly hear accounts of cities paying off 'barbarians' rather than resisting.

This historiographical debate is still going on currently, as seen in the Cambridge Ancient History volume XIII and XIV – vol. XIV, published in 2000, has an extremely cautious conclusion: ‘it is largely a matter of where one places the emphasis and why…’ In the 2000s, Peter Heather have returned to the idea of a 'decline', leading to a Counter Reformation led by Ward-Perkins (Someone should write a history of the historiography of Latter Roman history...). As recently as 2016, Harris' Roman Power: A Thousand Years of Empire (2016) has returned to the idea the Christianity played a part in the 'decline' of Rome.

But regardless of their views, both sides agree that:

'These facts (the survival of the east) are important, for they demonstrate that the empire did not, as some modern historians have suggested, totter into its grave from senile decay, impelled by a gentle push from the barbarians. Most of the internal weaknesses which historians stress were common to both halves of the empire … directly or indirectly, it may be plausibly argued, barbarian attacks probably played a major part in the fall of the West.” (Jones 1964: 1027)

'The internal weaknesses of the empire cannot have been a major factor in its decline.' (Jones 1964: 1068)

Besides sources mentioned above, further sources divided into the two schools:

'Decline' School:

Harris, W. V. Roman Power: A Thousand Years of Empire, (Cambridge, 2016).

Heather, P., Goths and Romans, 332-489, (Oxford, 1991).

Heather, P., The Fall of the Roman Empire: A New History, (London, 2005).

Transformation' School:

Cameron, A., The Later Roman Empire, AD 284-430, (Cambridge, Mass., 1993).

Drinkwater, J., and Elton, H. (edd.), Fifth-century Gaul: A Crisis of Identity?, (Cambridge, 1992).

Goffart, W., Barbarians and Romans, A. D. 418-584: The Techniques of Accommodation, (Princeton, 1980).

Goffart, W., Barbarian Tides: The Migration Age and the Later Roman Empire, (Philadelphia, 2006).

Halsall, G., Barbarian Migrations and the Roman West, 376-568, (Cambridge, 2007).

Ward-Perkins, B., The Fall of Rome and the End of Civilization (Oxford, 2005).

Wickham, C., Framing the Early Middle Ages: Europe and the Mediterranean, 400-800, (Oxford, 2005).

I could raise some disagreements about the later Roman Empire, but I'm not expert enough.

Noting previous replies is not intended to discourage further questions and rebuttals.

• FAQ section: The Fall of the Western Roman Empire, but no specific articles.
• FAQ section: Historians' views of Edward Gibbon's "Decline and Fall of the Roman Empire": no specific article titles mention Christianity, but it's touched on in several articles there.
• What did Edward Gibbon's contemporaries think of his claim that Christianity contributed to the fall of the Roman Empire?, by someone deleted.