How the English nobility felt about their longbowmen can mostly only be inferred from how they were treated rather than specific statements made about them by nobles.
The actions of both King Edward III and his son Edward the Black Prince would indicate that longbowmen were highly valued, though this does not automatically mean respect. During his reign, Edward III pardoned thousands of longbowmen for often violent crimes. Significantly, many of these pardons were at the request of senior nobles, making use of the king's patronage.
In his 2015 master's thesis The Archer’s Tale: An Examination of English Archers during the Hundred Years War and their Impact on Warfare and Society (pdf), Stephen Scott Taliaferro notes that:
The 1360 pardons list also records the names of the captains who
requested the pardon on behalf of their retainers. These men
represented the military elite of England and included the Prince of
Wales, John of Gaunt, the Duke of Lancaster, the Earl of Northampton,
and the Earl of Warwick among others. That such powerful men would
request pardons from the king on behalf of archers, demonstrates the
respect, or at least the desperation at the time to obtain that skill
This does not mean, of course, that these nobles suddenly regarded all archers as their social equals, but it does mean that longbowmen were a long way from being nameless cannon-fodder.
The Black Prince had his own elite guard of archers, and the Prince went to the time and trouble to sign a special pass for an archer by the name of William Jauderel, a descendant of Peter Jauderel who had served under Edward I. The pass reads:
Know all that we, the Prince of Wales, have given leave on the day of
the date of this instrument, to William Jauderel, one of our archers,
to go to England. In witness of this we have caused our seal to be
placed on this bill. Given at Bordeaux 16th of December, in the year
of grace 1355.
Source: Robert Hardy, Longbow: A Social and Military History
For a soldier to be named is a step up; to merit the personal attention of the then heir to throne signifies a not insignificant degree of regard. Also worth noting are the careers of the likes of Robert Knolles and John Hawkwood; both were archers who were knighted. Hawkwood even took as his second wife the illegitimate daughter Bernabò Visconti, Lord of Milan.
Perhaps the closest we have to contemporary spoken opinion (as suggested by Mark C. Wallace's comment) comes from Geoffrey the Baker's Chronicon Angliae temporibus Edwardi II et Edwardi III which records a speech supposedly given by the Black Prince to his archers before the Battle of Poitiers in September 1356.
Geoffrey the Baker was not actually at Poitiers but he was "able to describe [it] in great detail, apparently from what he was told by the prince's commanders."
You have often given me good proof of your courage and your loyalty.
In many fierce tempests of war you have shown that you are not
degenerate sons, but of the same blood as those men under the
leadership of my father and my ancestors as kings of England found no
task impossible, no place forbiddingly impassable, no mountain too
high to climb, no tower too strong to capture, no army unbeatable, no
armed enemy formidable.
Source: The Chronicle of Geoffrey Le Baker of Swinbrook (trans: David Preest)
There is undoubtedly a certain amount of pre-battle hype here, and Geoffrey continues with the inevitable appeal to "honor and love of your country" and the prospect of "spoils", and then
So follow the standards, with mind and body concentrated on the
commands of your leaders, so that, if life and triumph comes our way,
we may continue in that firm friendship which always wants the same
and scorns the same. But if jealous fortune, which
God forbid, should propel us down the final road of all flesh during the task which
lies before us, it is not imprisonments awaiting the wicked which will dishonor
your names. No, all of you together with me and with these noblemen, my
companions, would drink from the same cup.
This passage is quoted by Taliaferro. He prefaces the cited text with
Despite the obvious threat to the military supremacy of the
aristocracy that archers posed, the English military community evolved
out of necessity to include archers. As archers became the lynchpin of
the English tactical system, the attitudes of the English aristocracy
toward archers softened.