Otto Wagner, the father of modern Viennese architecture,

It is (A).

Anand I would appreciate comments from your side.

(Also as it is a reference to a "thing" and not to a person, so shouldn't "which" be used rather than "that")

The stem has an adjective phrase "the father of modern viennese architecture". This phrase can be moved around as follows without affecting the meaning of the sentence. Once the phrase is moved the commas also move with it.

the father of modern Viennese architecture,
Otto Wagner designed buildings that earned him an international
reputation and influenced an entire generation of artists and architects.

Now try to replace the underlined piece with answer choices

B) replacing this will have "Otto Wagner who designed blah blah"
The "who" is redundant and unncessary.
C) It changes the meaning because Otto Wagner earned the reputation because of the buildings he designed and not that the designs earned a reputation for him
D) "which" and "that" can both refer to anything other than humans.
"which" is generally used for non restrictive clauses and "that" is used for restrictive clauses. "that" is apropriate here.
E) It lookes like he earned reputation while designing the buildings. This is awkward

Outstanding explanation anandnk,

Thanks, Dharmin