b.bef 24 Oct 1600 Stepney, Middlesex, England
d.bef 12 May 1667 Bridgewater, Plymouth, Massachusetts, United States
Facts and Events
There is 1 vital record available on MyHeritage for William Bassett, including birth records, marriage records, and death records. Vital records are historical records that are typically recorded around the actual time of the event, which means they are likely accurate. Vital records include information like the event date and place, and the person's occupation and residence. Vital records also often include information about the person's relatives. For example, birth and marriage records include names of parents and divorce records list the names of children.
For a fairly comprehensive treatment, see William Bassett 1595??-1667
He appears on lists of Freeman in 1633 and 1637 at Plymouth, in 1639 at Duxbury, and in a list at Bridgewater probably from 1658. 
In the 1623 Plymouth land division "William Bassite" received two acres as a passenger on the Fortune in 1621.  The 1627 Plymouth division of cattle included William Basset, Elizabeth Basset, William Basset Jr. and Elizabeth Basset Jr. 
On 3 June 1652 William Bassett of Duxbury gave to "his son-in-law Leiftenant Perigrine White" forty acres of upland with the meadow adjoining.  On 16 June 1656 "William Bassett Senior of Duxburrow now living at Bridgewater" made a deed of gift of his Marshfield lands to his "two sons there living viz: Perigrine White and Nathaniell Bassett." 
On 3 April 1667 William Bassett Senior made a nuncupative will, bequeathing the movables to his wife, and the house and land to her during her life, after which it was to go to his son William's son, and bequeathing his tools to his son Joseph, and "being demanded about his books which he formerly took care about, answered he could not now do it." His inventory was taken 12 May 1667 and totalled £123 2s. 6d. (which included no land, but did include his blacksmith's tools and more than twenty books). On 5 June 1667 letters of administration were granted to William Bassett Jr. on the estate of William Bassett Sr. deceased. Two years later, "William Bassett of Sandwich ... the eldest son and heir of William Bassett sometimes inhabitant of ... Bridgewater ... now deceased" confirmed on 2 June 1669 to "Joseph Bassett of Bridgewater my youngest brother" land in Bridgewater granted him by his father in his lifetime but not legally confirmed.
Wives and Parents
William was married before the 1623 land division to a woman named Elizabeth (as listed in the 1627 cattle division) who most likely came on the Fortune as well, as his wife or as a single woman.
William married second before 12 December 1664 Mary (Tilden) Lapham, daughter of Nathaniel Tilden and widow of Thomas Lapham.
"In 1611 a William Bassett, formerly of Sandwich in England, widower of Cecily Light, was twice betrothed at Leiden in Holland. His first bride-to-be died, but he succeeded the second time. Some have held that this was the man who came to Plymouth, but this seems unlikely given the ten-year gap before the arrival in Plymouth in 1621, and the lack of evidence for children of the Plymouth man born before that date, assuming that he had been married at least twice before. It is also possible that the William Bassett of Leiden in 1611 was the father of the immigrant to Plymouth in 1621, but there is no evidence directly favoring this hypothesis. (See discussion in Stratton 242-43.)"S1
Candidate parents given by various theories:
Note that Great Migration Begins says that nothing is known about his origins. Further, quoting the article cited above: "He may conceivably have been a son of the Pilgrim William Bassett of Leyden, Holland, but Robert Charles Anderson in The Great Migration Begins (page 127[-130]) says there’s no evidence." The parents listed here are speculative.