IMO A. Modifiers look correct in A.

I feel its B...

(B) Not long after the commencement of the decade of the 1930s, baritone singers such as Bing Crosby and also Russ Columbo decided to contribute in
"and also" incorrect, and "decided to contribute in" and "contributed to" are very different in meaning.

(C) Not long after the 1930s commenced, baritone singers like Bing Crosby and Russ Columbo contributed to
"like" is incorrect - test of like/such as usage.

(D) Not long after the beginning of the 1930s commencement, baritone singers like Bing Crosby and Russ Columbo had contributed to
same with above.

(E) It was not long after the 1930s commenced that baritone singers such as Bing Crosby and Russ Columbo had contributed in
"had contributed" is incorrect tense.

A - Perfect.

B - 1. decade of the 1930s - redundant
2. the commencement of the decade of the 1930s - wordy
3. such as Bing Crosby and also Russ Columbo - also not required.
4. decided to contribute - shift in meaning. This would mean that after the 1930s commenced, Bing and Russ decided to contribute to this new type of singing. So, they actually may have contributed much later, say in the 1940s. This unnecessarily creates new unwarranted ideas.
5. contribute in - unidiomatic - contribute to is right.

C - baritone singers like Bing Crosby and Russ Columbo - Incorrect use of like. Like means similar to. This would mean that not Bing and Russ but singers similar to them contributed to crooning. This changes meaning.

D - 1. beginning of the 1930s commencement - redundant - beginning and commencement mean the same.
2. baritone singers like Bing Crosby and Russ Columbo - same problem as in C.

E - 1. Bing Crosby and Russ Columbo had contributed in - Wrong tense - use of perfect form had is unwarranted.
2. contributed in - unidiomatic - contributed to is right.

The correct idiom is " contributed to "

So B and E out

In D beginning and commencement are used together and beginning is useless here.

In C use of like is wrong.

A wins