Technically, Pakistan's blasphemy laws are inherited from the British. In the early days of the Raj, the British Parliament appointed a commission chaired by Lord Macaulay to create a comprehensive penal code for British India. Chapter XXV of the resulting Indian Penal Code, adopted in 1860, covers offenses related to religion. The amended IPC was adopted as the Pakistan Penal Code upon independence; it also underlies the basic criminal code of India, Bangladesh, Malaysia, and elsewhere.
Two notes must be made, however. First, the nature of Pakistan— an Islamic state or a secular state for Muslims?— has been prominent in domestic political debate since its founding, along with the particular expression and enforcement of its Islamic identity. Indeed, even the constitution, architected in 1973 by the relatively secular-oriented Zulfikar Ali Bhutto, calls for the Islamization of the law, and as prime minister, Bhutto made Islamic studies mandatory in schools and backed a ban on the sale and consumption of alcohol by Muslims.
Second, the Islamization of Pakistani law and society along Saudi lines is attributable to the government of Gen. Muhammad Zia-ul-Haq. By all accounts, Zia was personally pious and saw Islam as Pakistan's salvation. Saudi Arabia provided generous financial support to Pakistan both directly to the government and indirectly, for example by funding religious schools. Saudi interest in Pakistan increased after revolution in neighboring Iran brought a Shi'a government to power. Zia and the elites of his era were thus close to Saudi Arabia, and no doubt took Saudi advice to heart. Pakistan even sent troops to Saudi Arabia in response to unrest in the south, and hundreds of thousands of Pakistanis to this day find work in Saudi Arabia.
From Zia's coup in 1977 to his death in 1988, Pakistan adopted increasingly strict laws, such as the Hudood Ordinance and the Zina Ordinance. The blasphemy laws were extended with Islam-specific provisions and the punishments made much more severe in 1980, 1982 and especially 1986. Few were actually prosecuted under the Hudooda and Zina provisions, but that cannot be said of the new blasphemy laws. Fewer than 10 blasphemy cases had been filed from Pakistan's creation to the 1986 ordinance; since then, there have been thousands.
Wikipedia offers an overview of blasphemy laws around the world, with the usual caveats about using Wikipedia applicable.
Incidentally there are many critics within Pakistan, including conservative mullahs, who believe the laws are used to settle personal scores and persecute political opponents rather than enforce public morality. Most charges are made against Muslims, after all, not members of religious minorities. But according to the BBC, many Pakistanis mistakenly believe such laws are taken from the Koran. Given the laws' popular support, reform will be low on the agenda for Pakistan's political parties.