The Quran prescribes no punishment for blasphemy. Rather it teaches Muslims should not blaspheme against the beliefs of others and should respond to those who blaspheme against their own beliefs by leaving any such conversation. 

When you hear the Signs of Allah being denied and mocked at, sit not with them until they engage in a talk other than that.

- Quran 4:141

The Quran's only instruction regarding blasphemers is that Muslims should not sit with them while they continue to blaspheme. 

The fourth Caliph of the Ahmadiyya Muslim Community has commented on this incident:

Yet there is a case of blasphemy so serious that it is recorded in the Holy Quran itself, which speaks of Abdullah bin Ubayy bin Salul, known in the history of Islam as the Chief of Hypocrites. Once returning from an expedition, Abdullah bin Ubayy declared in the company of others that the moment they returned to Medina, the noblest would expel the meanest among the Medinites...

Everyone understood the implied insult to the Holy Prophet^sa. They were seething with indignation and rage to the extent that, if permitted, they would have most certainly put Abdullah bin Ubayy to the sword. It is reported authentically that tempers were running so high at this incident that no less a person than the son of Abdullah bin Ubayy himself approached the Holy Founder^sa of Islam seeking permission to kill his father with his own hands. The son argued that if anyone else did so, he might later on, in ignorance, entertain the thought of revenge against his father’s killer. For centuries, the Arabs were accustomed to take revenge at even the smallest insult hurled at them or their close relative. Perhaps, this custom was what the son had in mind. But the Holy Prophet^sa refused to grant his request nor did he permit anyone else from among his Companions to punish the hypocrite, Abdullah bin Ubayy, in any manner whatsoever.

Having returned to Medina after the expedition, Abdullah bin Ubayy continued to live in peace. When he died at last, a natural death, of course, to the surprise of everyone, the Holy Prophet^sa gave Abdullah’s son his own shirt so that he could enshroud his father’s body for burial - a singular act of blessing, indeed, which must have left the other Companions yearning to barter it from the son at the cost of all their possessions. Not only that, the Holy Prophet^sa decided to lead his funeral prayer. This decision must have deeply disturbed many of his Companions who could never forgive Abdullah for the grievous offence mentioned above. Yet, it fell to the lot of Umar^ra, who later succeeded the Prophet^sa as the Second Caliph to give voice to their suppressed uneasiness.

It is reported that as the Holy Prophet^sa was proceeding to the funeral, Umar^ra suddenly stepped forward and stood in the way begging the Prophet^sa to change his decision. In doing so, Umar^ra reminded the Prophet^sa of the verse of the Holy Quran in which reference is made to some known hypocrite on whose behalf intercession would not be accepted even if the Holy Prophet^sa prayed for him seventy times. Incidentally, the number seventy should not be taken too literally because, according to Arab usage, it was only employed to indicate a large number.

However, the Holy Prophet^sa smiled and responded: Stand aside, Umar^ra. I know better. If I know God would not forgive him even if I seek forgiveness seventy times, I would seek forgiveness for him more than seventy times. The Prophet^sa then led the funeral prayer.

This is a fitting rebuttal to those who are crying themselves hoarse in demanding death for the blasphemer who dares to insult the Holy Founder^sa of Islam and nothing but death. Such a religion must have a claim to establish inter-religious peace in the world. 

- Hazrat Mirza Tahir Ahmad, Islam's Response to Contemporary Issues