Specifically before the Conquest?
Before we answer this question, we need to clarify William the Conqueror's birth status. The controversy, and the origin of his nickname William the Bastard, derived from his parents being unmarried at birth. However, although his mother, Herleva, was a Tanner's daughter, his father was of course Robert I, the sixth Duke of Normandy. Also, it must be noted, he was Edward the Confessor's mother's great nephew (Edward being the reigning King of England at the time of William's birth). So the qualms about his birth status essentially come from William being born out of wedlock, rather than his lineage (which was stellar for the period).
It is unclear if William would have been supplanted in the ducal succession if Robert had had a legitimate son. However, sundry earlier dukes had been illegitimate, and William's association with his father on ducal charters appears to indicate that William was considered Robert's most likely heir. The Lord convened a council in January 1035 and had the assembled Norman magnates swear fealty to William as his heir. The notion of bastardy held less sway back in the 11th Century, when society was still relatively simple compared to the later medieval period, and opportunism and pragmatism held precedence over purity of descent.
This can perhaps be attributed to the extent of Christian and Papal permeation into French and English society at the time (both had previously had a pagan history, though France less so given the heavier Roman influence). The idea of Papal supremacy and Catholicism, and thus Christian ideas about bastardy, were still new in France. This meant that many Lords did not hold the same sort of 'snobbery' as later rulers.
Indeed, the concept of French and English society is perhaps inaccurate, given that England and France were just a collection of ever-changing kingdoms (Even King Cnut's (a variant in spelling of Canute in English or Knud in modern danish) jurisdiction was only a part of modern England (Mercia, Northumbria, Wessex). The image we may have of daily, gossip-filled, court politics just didn't exist this early. In fact, the knowledge that William was a Bastard would take many years to travel. Significantly, though, the fearless Danes now populated and dominated much of Northern France and England, dominating the continent with their long boats; they were less concerned about birth status and more about wealth and military might.
Also, at the time, French lords would have a high degree of autonomy. William ruled a dukedom smaller than Yorkshire, for which he paid homage to the French King. His own barons were fairly autonomous - only bound to him by a reciprocal agreement of war loyalty for the land. Travel across land was still an ordeal, more so across water. So, again, William would have led a life fairly detached from other lords and noblemen. Allied with the fact that his succession had the support of Henry I (French King), the issue would not be a major one among nobles.
In conclusion, on balance, at the start of his reign as duke, I think the thing that would have most influenced his treatment would be his lowly age. He was only seven when he supplanted his father! In this opportunist period, material wealth and military hard-power would have dictated the terms of engagement between nobles much more than circumstances of birth.
Sources:
William the Bastard had a difficult time securing Normandy after the untimely death of his father, Duke Robert, on pilgrimage in 1035. Indeed, Duke William's minority was one of the most difficult times of his entire career. He had to fight off rebellious family members who used his bastardy as justification for revolt. For example, in one incident, a town Duke William was besieging taunted him by hanging animal hides on the walls (in reference to his maternal grandfather, who was a tanner). In response, Duke William rounded up a number of captives from the town and brutally chopped off their feet and hands. He was certainly sensitive to this sort of mocking.
After the battle of Battle of Val-ès-Dunes in 1047, the tide turned in Duke William's favor and he began the long process of reclaiming his duchy. And once he asserted his control of Normandy, most of vassals did not express problems with his bastardy. Outside of Normandy, however, this was the rule as well; would you want to anger one of your most fearsome neighbors by insulting his birth? One incident does stand out, though. When Duke William asked for the hand in marriage of Matilda, daughter of the Count of Flanders, she allegedly spurned him. Duke William, then, allegedly rode to Flanders, pulled her off her horse by her hair and demanded she marry him, to which Matilda agreed. By all accounts the couple enjoyed a happy marriage that lasted for decades.
Tl;dr His bastardy was used as a justification for revolt by Norman vassals at first. Once he became powerful, that wore off real fast as William the Bastard reacted in typical bastard fashion and inflicted terrible violence on anybody who dissed him.
Sources:
David C. Douglas, "William the Conqueror: The Norman Impact Upon England," 1967
Richard Huscroft, "The Norman Conquest: A New Introduction," 2009
Edit: Yes, Duke Robert died on pilgrimage, not crusade. My mistake; always proofread your work, folks!