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Henschen was born at Venray, Limburg, in the Low countries. He was the son of Henry Henschen, a cloth merchant, and Sibylla Pauwels. He studied the humanities at the Jesuit college of Bois-le-Duc (today the town of 's-Hertogenbosch) and entered the novitiate of the Society of Jesus at Mechlin on 22 October 1619. He taught successively Greek, poetry and rhetoric at Bergues, Bailleul, Ypres, and Ghent. He was ordained a priest on 16 April 1634, sent to the professed house at Antwerp the following year, and admitted to the profession of the four Jesuit vows on 12 May 1636.
From the time of his arrival in the city he was associated as collaborator with his fellow Jesuit, Jean Bolland, who was then preparing the first volumes of the Acta Sanctorum. It was Henschen who, by his commentary on the Acts of St. Amand, suggested to Bolland the course to follow, and gave to the work undertaken by his mentor its definitive form.
At Bolland's direction, Henschen journeyed in company with Daniel van Papenbroek, to Italy, France, and Germany (22 July 1660 – 21 December 1662) to collect ancient documents for their studies. Upon their return, they learned that Bolland had died, at which point he and Papenbroek began to lead the project. He was the first librarian of the Museum Bollandianum at Antwerp.
Henschen died at Antwerp, aged 80, in 1681.
Henschen collaborated on the volumes for January, February, March, and April, and on the first six volumes for May, that is on seventeen volumes of the Acta Sanctorum. Several of his posthumous commentaries appeared in the succeeding volumes. A list of some other works from his pen will be found in Augustin de Backer's Bibliothèque des escrivains de la Compagnie de Jésus.
- Daniel van Papenbroek, De vitâ, operibus, et virtutibus God. Henschenii in Acta Sanctorum, VII, May
- Joannes Joseph Habets, Godfried Henschenius medestichter der Acta Sanctorum (Maastricht, 1868).