The Bassett Family Association does not recognize any origins or parents (including John Bassett and Margery Bass, suggesting that it has not yet been proved that William Bassett of New Haven was the child baptized 30 May 1624 in St. Martin, Dorking, Surrey, England.
It is stated in many internet places that William Bassett arrived in Boston, Massachusetts in 1635 in the ship Truelove, from London, England, in company with Rev. Peter Hobart, John Cooper Sr., John Cooper Jr., William Ives, Abraham Dickerman, and others. William married Hannah (Dickerman) Ives, the widow of William Ives, daughter of Thomas and Eleanor (Whittington) Dickerman, on Tuesday, 7 Nov 1648 in New Haven, Connecticut.
The located listings for the ship "Truelove" in 1635 (Sept) do not contain any of the names listed, making William's presence aboard suspect and certainly unproven.
William did marry the widow of William Ives, but her identification as a Dickerman is questionable.
Spouse Hannah Dickerman ?
Hannah, wife of William Ives and wife of William Bassett, has been called Hannah (Dickerman) in some secondary sources, including the Bassett Family Association. None of these presents us with evidence of this surname. However, when William Bassett died, he called Abraham Dickerman and John Cooper "his brothers", to be overseers. Of course, this does not mean that either man is his actual brother or brother-in-law. John Cooper, Abraham Dickerman and William Bassett had lived together in a small community for years. In addition, William's son Samuel had recently married Mary Dickerman, who was the child of Abraham Dickerman and the grandchild of John Cooper. There were no known children of Abraham Dickerman called Hannah.
William Bassett who ultimately resided in New Haven, Connecticut, was born about 1623 in England. This would make him 25 when he married in 1648.
William took the New Haven Oath of Fidelity, 1 July 1644, in the first group of men to do so.
It is not unusual in the early New Haven records for men to receive fines for various infringements of the rules. William Bassett is no exception. The first was a 6 d fine on 7 March 1643/4 Will Bassett and others were fined for "foole guns." He had the usual litany of misdemeanors: late for coming to the Lords day 1 May 1644; defective gun 7 July 1646; 4 July 1648 coming to Sabbath without his armes; absense from training 7 Nov 1648 (he had hay cut that needed to be picked up, so was excused without a fine; 6 Mar 1648-absent two days of court, (on one he was keeping cows, another his wife was sick, which the court passed) on these; for not bringing his armes on the Lords day; Oct 1649 for coming late to watch 12d.
7 Nov 1648: "The will of William Ives deceased was presented in court, made the 3th of Aprill, 1648,
"William Basset whoe is neare the mariage, (they being contracted,) of ye widdowe, was called to put in securitie to ye court for the estate, that the chilldren of William Ives maye have their portions duely pd, according to ye generall courts order, but he desired respite till ye next court wch the court granted."
He was called to court 5 Dec 1648 to put in security for the portions of his wife's children but it was respited and then again on 6 Feb 1648 when he engaged the whole estate which was left him by William Ives "& will not alter any of it till he acquaint the court wht it & put in as good an estate as he shall dispose of." 
William died in 29 Aug 1684, in New Haven, Connecticut.
In his will dated Jan 1 1679 and witnessed by Thomas Yale, Sr. and James Clarke, he makes bequests to daughters Bia [Abiah] Bassett, Phebe Rose [step-daughter] and Hannah Parker, and to son Joseph Ives [step-son]. He calls Abraham Dickerman and John Cooper his brothers, to be overseers. The inventory amounting to £50:10:00 was taken Sept 1684. Abraham Dickerman and John Cooper had recently become his in-laws as his son, Samuel, married Mary Dickerman, daughter of Abraham and granddaughter of John Cooper, on June 21, 1677.
"For some reason, not known, this will leaves out his other children and step children: John Bassett, Samuel Bassett, Martha (Ives) Beach, and John Ives. John and Samuel Bassett were living when he wrote his will as was John Ives (although John Ives died in 1682, two years before his step father passed on). The date of Martha Ives Beach’s death is not known. After William Bassett died on August 29, 1684*, his estate was inventoried by John Winston and Joseph Mansfield who reported a value of 50 pds, 10 shillings on Sept. 1684."
The second volume of New Haven Colony records does not contain his name
He is mentioned in the Town records not included here yet.
↑ Hotten, John Camden (editor). The Original Lists of Persons of Quality: Emigrants, Religious Exiles, Political Rebels, Serving Men Sold for a Term of Years, Apprentices, Children Stolen, Maidens Pressed, and Others, who Went from Great Britain to the American Plantations, 1600-1700. London: John Camden Hotten, 1874 at Archive
↑ "Abstracts of the Early Probate Records of New Haven, Book I, Part I, 1647-1687." New England Historical and Genealogical Register 81:122/3. Boston: NEHGS, 1927. p 121-135
↑ Jacobus, Donald Lines (compiler). Families of Ancient New Haven, Vol I-VIII. and Index Vol IX New Haven: 1931. Reprint, Baltimore: Genealogical Publishing Co., Inc., 1974, 1981, 1997. Originally published as New Haven Genealogical Magazine, Volumes I-VIII. Rome, NY and New Haven, CT 1922-1932.
I didn't start out to work on this, (I was making sure that Hannah wasn't a Dickerman). As I was trying to sort her, I found the interesting things about Bassett. I will try to get back to William in not too long.