Is this sentence punctuated correctly?
The wanderer found no reason to be wary of him, but nonetheless the proud old man replied, “No need, I am quite all right,” as he plodded forward through the street.
I would place a comma after word "nonetheless," since a definite pause would help the reading of the sentence. Otherwise, the sentence looks good as far as punctuation.
The sentence is “correct” as given, but there are some variations that might be stronger. First, [need, I am… ] could be [need; I am..] because the implied “There is” makes it two coordinate clauses [There is no need;] and [I am quite all right] that can be best separated by a semicolon. Also there is some question about the necessity for a comma before the quotation. Finally, the punctuation at the end of the quotation could be an exclamation mark [!] and it would be inside the quotation as part of the utterance. So a somewhat smoother solution would be:
The wanderer found no reason to be wary of him, but nonetheless the proud old man replied “No need; I am quite all right!” as he plodded forward through the street.
Looks good. I would add a comma after "nonetheless." This is a preference.