There are two major arguments here:

-The British didn't directly prevent the violence. Both police and military units were mostly allocated to prevent attacks on British enclaves and personnel, as the British expected that they would become targets of violence themselves; this wholly failed to manifest (mostly because the pro-India and anti-British groups were fairly non-violent to begin with, and also attaining their great goal of independence; what use attacking an Englishman?) It's also fair to say that the rowdies found much easier and more numerous targets. Presumably if the British had focused less on self-protection against a mostly non-existent threat, more resources would have been available to prevent the sectarian flare-ups that eventually touched off the general violence.

-On the other hand, the British are also roundly (and more justly) blamed for having had a major hand in creating the situation around which the violence was based; in other words, for their colonial history of valuing different castes and religious groups differently and for having agreed to Partition in the first place.

It's undeniable that the British simply didn't see "Indian" as a nationality around which the Indian people could coalesce. Instead they saw a lot of little nationalities, castes, etc. This isn't to say that the British ignored caste - they were perfectly willing to take advantage of it to the extent that it provided them with certain groups that were happy to provide military service, for example. (Indeed, the British were of the opinion that the non-military castes simply weren't fit for such service in the first place; certainly not an enlightened view of race/culture/caste at all, only partially excused by the fact that it was originally an Indian distinction...)

This logic is pretty straightforward. The British practice of dividing India into caste/religious/other communities worked actively against the creation of a pan-Indian identity. This opened up the possibility for the Muslim League to advance its demands for Partition (or if you're more charitable, you can phrase that as "gave the Muslims in the League reason to doubt that a united India would treat them well as a minority"). Further, the British actually agreed to partition the country - it could have handed over control of the entire country to Congress, which would have left them a nasty ethnic problem, but one they may have been able to deal with without widespread violence, since the trigger of Partition and mass migration would not have been there.

Are these unfair charges? Perhaps somewhat? The British honestly had worries for their security and allocated their resources accordingly; they honestly believed that there were significant differences between the various Indian sub-communities that would prevent them from operating as a unified whole (certainly Churchill was of this opinion); they didn't have any reason to believe that Partition would necessarily erupt into immediate violence, or at any rate that violence was inevitable under Partition and avoidable otherwise.