Basset is a minor noble under Somerset. He is loyal to his lord, and wears a red rose to show his nobility. He and Vernon argue about the validity of York’s claim to the Dukedom, and eventually beseech the King for permission to fight, which he denies.
This may not be the Basset Shakespeare had in mind when he penned 1 Henry 6, but it is the only one we have a record of from the correct time period.
The following information is quoted from the Oxford Dictionary of National Biography, written by Anne Curry.
Basset, Peter (fl. 1415–1437), soldier and historian, was co-author of an incomplete account in French of English activities in France between the capture of Harfleur in 1415 and the raising of the siege of Orléans in May 1429. Only one manuscript of this work is known, in London, College of Arms, MS 9, folios xxxir–lxvv, written in an early fifteenth-century hand but bound up with papers of a later date. According to a Latin preface inserted by William Worcester, the work was compiled for Sir John Fastolf in the year of his death (1459) by three men, Peter Basset, Christopher Hanson de patria almayn, who had served under Thomas Beaufort, duke of Exeter (d. 1427), and Luke Nanton, native of ‘Pekyant’, one of Sir John Fastolf's clerks, ‘by the diligence’ of Worcester himself. Worcester's hand can be seen in corrections and glosses. The exact contribution of each author is unknown, but the chronicle is a bare narrative embodying extensive lists, both for the English and French, of those present, taken prisoner or killed in engagements, or appointed to captaincies. Edward Hall (d. 1547) drew extensively on the text and lists for his own chronicle. In his preface Hall names a Ihon Basset as one of the English writers he had consulted, but in his account of the death of Henry V states that Peter Basset ‘which at the time of his death was his chamberlain’ was the source of the information that the king had died of ‘a plurisis’ (Hall's Chronicle, 113). In 1557 John Bale ascribed to Peter Basset a work in English entitled Acta regis Henrici quinti; he also claimed that Basset had been Henry's chamberlain, and noted the same statement concerning the cause of death, a statement that Bale seems to have Basset as making in 1430. There is no contemporary corroboration that Basset was Henry V's chamberlain, nor has any such English work been identified.
Basset is described in Worcester's preface to the chronicle in College of Arms, MS 9, as an esquire of the English nation who had served in arms in France for thirty-five years under Henry V, John, duke of Bedford (d. 1435), and several other royal lieutenants. Several Peter Bassets occur in the records of English armies in France in the years after 1415. But given the link with Fastolf, it seems highly likely that the historian was the Peter Basset who stood bail in 1426 for the procureur of Pirmil when the latter had been flung into prison by the captain of Isle-sous-Brûlon in Maine, which lay near to Hanson's garrison of Ste Suzanne, and who was one of the men sent to reinforce Fastolf's garrison of Alençon in October 1429. He is later found among reinforcements at Essay (March 1431), and served at the siege of St Celerin in 1432, rejoining the garrison of Alençon in January 1434 and remaining there until at least December 1437. No record has been found of his service in Fastolf's English administration, whereas Hanson and Nanton were certainly in Sir John's employ until the latter's death.