To be clear, I am not asking why Aragorn and the Dunedain are treated as suspicious, but why the nickname 'Strider' (and Longshanks) is implied to have an insulting undertone, by Bill Ferny and Butterbur, (beyond the unintended disrespect of not addressing him by his proper title as Aragorn, Son of Arathorn, Heir to the Throne of Gondor) which I can understand his need to conceal. The name (stick-at-nought) Strider seems to be a slur in and of itself - is there a meaning or reason behind the word that makes it offensive?
"Strider" wasn't an insulting name
Aragorn was known to the people of Bree as a stern outsider who kept to himself. That was enough for him to be looked on with suspicion even by good people like Butterbur.
When Bill Ferny speaks insultingly of him, he says
'‘I suppose you know who you’ve taken up with? That’s Stick-at-naught Strider, that is! Though I’ve heard other names not so pretty.’
The Lord of the Rings Book One, Chapter 11: A Knife in the Dark
It's not "Strider" that is insulting, it's "Stick-at-naught" (stop at nothing), in other words he is saying that Strider has no scruples.
The only other comment on the nature of the name "Strider" I can think of is when Pippin first sees Aragorn after the Battle of the Pelennor Fields. Pippin greats him as "Strider" and Imrahil says to Éomer
‘Is it thus that we speak to our kings? Yet maybe he will wear his crown in some other name!’
And Aragorn hearing him, turned and said: ‘Verily, for in the high tongue of old I am Elessar, the Elfstone, and Envinyatar, the Renewer’: and he lifted from his breast the green stone that lay there. ‘But Strider shall be the name of my house, if that be ever established. In the high tongue it will not sound so ill, and Telcontar I will be and all the heirs of my body.’
The Lord of the Rings Book Five, Chapter 8: The Houses of Healing
In this case, I don't think Imrahil considered "Strider" to be insulting, I think he just didn't think it was grand enough for a King.
This answer was written when the question was still unclear, please bear that in mind. Blackwood's answer, however, is a more in-depth answer to the question in its unraveled state.
Only in Bree is the name ‘Strider’ offensive.
Firstly, The reason for them not calling him Aragorn, Son of Arathorn is because they are unaware of his heritage. Few people in the wild know of Aragorn's heritage. Aragorn would prefer to keep it that way, as this would be beneficial information to the Dark Lord.
On to the question at large, if I understood correctly you're asking about this exchange:
“Morning, Longshanks!’ he said. ‘Off early? Found some friends at last?’ Strider nodded, but did not answer. ‘Morning, my little friends!’ he said to the others. ‘I suppose you know who you’ve taken up with? That’s Stick-at-naught Strider, that is! Though I’ve heard other names not so pretty. Watch out tonight!”
Fellowship of the Ring - Book One: Chapter 10, Strider
Between Bill Ferny and the company of Strider and the Hobbits.
The reason for Bill Ferny using the name Strider, Longshanks and Stick-at-naught in an insulting manner is because he is an agent of evil, and aware of Strider's skill in the wild.
He attempts to make the Hobbits fear their trusty guide so that his "friends", the Black Riders, can come pick off Frodo in Bree, rather than have to try to chase a skilled Ranger through a forest he knows better than the back of his hand.
“They will know all the news now, for they have visited Bill Ferny; and probably that Southerner was a spy as well.”
As for Butterbur. He keeps his reservations because he doesn't like Strider's kind. Aragorn intentionally keeps himself to himself when in places like the Prancing Pony to prevent attention being drawn to himself. This enables him to appear and disappear as he pleases. As the innkeeper, Butterbur doesn't like the mysterious types, as they are most like to cause trouble.
“Well, you know your own business, maybe,’ said Mr. Butterbur, looking suspiciously at Strider. ‘But if I was in your plight, I wouldn’t take up with a Ranger.”
“At last Mr. Butterbur went out, with another doubtful look at Strider and a shake of his head.”