Oh Stannis’ opinion of the nobles is all over his conversations with Davos, and it’s consistently not very flattering.

Cut for length.

“The letter… What did your lords make of it, I wonder?”

Stannis
snorted. “Celtigar pronounced it admirable. If I showed him the
contents of my privy, he would declare that admirable as well. The
others bobbed their heads up and down like a flock of geese, all but
Velaryon, who said that steel would decide the matter, not words on
parchment. As if I had never suspected. The Others take my lords, I’ll
hear your views.”

“You have more to say about the letter. Well, get on with it. I did not
make you a knight so you could learn to mouth empty courtesies. I have
my lords for that. Say what you would say, Davos.”

 

So…flatterers and fools, neither of which Stannis suffers gladly. Stannis is very much aware of how proximity to the king is a way to access power in feudal politics and of how those around him would be looking to benefit from that proximity. He has a very personal experience with how courtiers and lords flock to those in power looking to get something out of them from his time in King’s Landing, especially, I reckon, during that period where he was heir presumptive to the throne before Joffrey’s birth. He sat on the small council for years listening to people who told Robert what he wanted to hear or openly gave counsel geared to advancing their own interests rather than that of the king’s or the realm’s. He saw the corruption permeating Robert’s court as a result, the way those in power exploited it, and how Robert enabled that, and he is adamantly against that. Stannis is so sensitive to any form of flattery coming from any lord or lady that he can not help but comment to Jon that Lyanna Mormont having a namesake in Lyanna Stark was to curry favor with Ned. So he is skeptic of lords because that tint of self-interest and pursuit of advancement in those around the king that he is so familiar with taints their counsel and the empty courtesies covers what they really think. But what Stannis is after, as he demonstrates with Davos, is someone who tells him the truth and gives him honest and good counsel, even when he does not want to hear it. 

To fan Stannis’ skepticism of the honesty and honor of the noble class, the mess with Renly then happens, further diminishing his view of them.

The storm lords will not rise for me. It seems they do not like me, and
the justice of my cause means nothing to them. The cravenly ones will
sit behind their walls waiting to see how the wind rises and who is
likely to triumph. The bold ones have already declared for Renly.

His contempt is highest of those who declared for Renly, including those who defected to his side the moment Renly fell. 

“Good men and true will fight for Joffrey, wrongly believing him the
true king. A northman might even say the same of Robb Stark. But these
lords who flocked to my brother’s banners knew him for a usurper. They
turned their backs on their rightful king for no better reason than
dreams of power and glory, and I have marked them for what they are. Pardoned them, yes. Forgiven. But not forgotten.”

And he is right to feel that way. Renly’s platform was “might makes right” and “I deserve to be king because I’m handsomer and more likable”; his kingship would have been catastrophic to the social structure of the realm. There was no cause there, only a transparent power grab wrapped in a bouquet of roses. But lords declared for him because it was more beneficial to their interests to have Renly on the throne. Even when some defected to Stannis, there was a lot of brashness and overtness in the parley with Cortnay Penrose, and in the general attitude of these lords that neither Stannis nor Davos trust their loyalty or counsel. And honestly, why should they? As Davos says, “last year they were Robert’s men. A moon ago they were Renly’s. This morning they are yours. Whose will they be on the morrow?“

That’s a very good point, Onion Knight. 

Speaking of Davos, the fact that Stannis singles out the onion knight as the one person who would steer him true really reflects his view of his other counselors as well, and neither is unearned. Davos is the only one consistently giving Stannis good counsel with the sole goal of serving him rather than currying favor or gaining titles. He is the one speaking for Ser Cortnay’s loyalty and advising they strike for King’s Landing while the others are enthusiastically harping on who gets to kill a man who was their ally literally just the day before, or suggesting they threaten his elderly father in front of him. He is the one arguing that the people of Claw Isle are innocent and the kin of those who died fighting for Stannis at that. He is the one putting his life on the line to save Edric Storm and reminding Stannis of where his true duty lies and that a king who does not protect his people is no king. His is a counsel that Stannis deservedly values over that of the chattering “magpies”.

“You have a passing clever father, Devan,” the king told the boy
standing by his elbow. “He makes me wish I had more smugglers in my
service. And fewer lords…”     

More smugglers, fewer lords. He likens their counsel to the braying of mules at one point, and he is utterly unimpressed by their bluster and haughtiness. His lords think they deserve to be heeded and respected simply because they are highborn, whereas Stannis is a man who looks to deeds as a base for judgement of character (”you were a hero and a smuggler”) Honestly, Stannis’ arc is one long “these lords suck. The lordly class sucks. What a bunch of sycophants. Why can’t y’all be Davos?” 

He explicitly makes his opinion of the current lordly class very clear when he is raising Davos to lordship.

 "To be lordly is to be false. I have learned that lesson hard. Now, kneel. Your king commands.”  

Yeah, that is not a man who thinks highly of the nobility at all, but rather one who is completely fed up with the corruption, sycophancy, dishonor and exploitation he has witnessed of the nobles that he wants to get rid of them. 

Stannis glowered up at Theon where he hung. “You are not the only
turncloak here, it would seem. Would that all the lords in the Seven
Kingdoms had but a single neck… ”

 

No, Stannis won’t actually kill them, but replace them? That he could do. “We will make new lords” comes in response to Davos telling him that the highborn lords will never obey him, a lowborn upjumped smuggler from Flea Bottom. And Stannis is so done with that, with highborn lords prancing around thinking the world of themselves and leading Stannis astray (see: the Florents.) while looking down on the one person who actually earned his place and who gives honest counsel and encourages Stannis’ better self with no ulterior motive or an eye towards advancement. Because the way the highborns deride Davos absolutely ties into Stannis’ view of the nobles.

And this is where the part about affinity for the downtrodden meets Stannis’ political platform. That part is less visible in the text due to Stannis’ aloofness but it rears its head with Proudwing, Davos, and Jon Snow. Stannis is an anomaly in Westeros in his choice of counselors: oh, he does have highborn counselors but the people whose words he regularly heeds? Davos Seaworth, Melisandre, and Jon Snow. A lowborn smuggler, a foreign sorceress and a bastard with claims of oathbreaking hanging over his head. This isn’t devoid of pragmatism, of course, since he needs them for various reasons but Davos and Jon, in particular, represent an actualization of Stannis’ proclamation of raising new lords. Westeros is very invested in keeping the status quo between the aristocracy and the peasantry and openly frowns upon on anything that threatens that hierarchy or the power of the nobles. Raising a Flea Bottom denizen to a lordship and a Handship defies that. Wanting to install a bastard with an unknown mother as the Warden of the North defies that. The dominant view among Stannis’ social class is that peasants are of no importance and that bastards are treasonous and faithless, but Stannis sees their merit and rewards them for it. His esteem is not reliant on birth but on the character of the person he is dealing with because he cares about substance first and foremost.