Introduced into the College of Engineering only five years ago, ergonomics have by now become an extremely popular specialization; this trend can be partially understood as reflecting safety's importance in the workplace.
A. have by now become an extremely popular specialization; this trend can be partially understood as reflecting safety's importance in the workplace
B. has by now becoming an extremely popular specialization; this trend can be partially understood as a reflection of safety's importance in the workplace
C.have by now becoming an extremely popular specialization; this can be partially understood as a reflection of the importance of safety in the workplace
D.having now become an extremely popular specialization; this can be partially understood as a reflection of the importance of workplace safety
E. has now become an extremely popular specialization; this trend can be partially understood as reflecting the importance of safety in the workplace

A - This answer choice is grammatically incorrect. The plural verb have...become does not agree with the singular subject ergonomics.
In addition, the phrase safety's importance is ungrammatical. Expressing possession using 's is only allowed with concrete (that is, tangible, or not abstract) nouns.

B - While this answer choice corrects the original Subject Verb Agreement mistake by changing the plural verb have to the singular verb has, it introduces another grammatical mistake by changing the V3 form become to the V+ing form becoming.
The verbal construction has...becoming is grammatically incorrect. It mistakenly mixes different parts of two different tense forms: have/has which is used to construct the Present Perfect (together with V3 forms), and the V+ing form which is used in the Progressive tense (with the verb to be).
In the current sentence the use of the Present Perfect (if correctly formed with has/have + V3) would be appropriate: ergonomics started to become popular since the specialization began at the college five years ago (in the past), and this trend is relevant/continues into the present.
In addition, the phrase safety's importance is ungrammatical.
C - This answer choice repeats the original mistake concerned with Subject Verb Agreement. In addition, this answer choice similarly to answer B introduces another grammatical mistake by changing the V3 form become to the V+ing form becoming.

The verbal construction have...becoming is grammatically incorrect. It mistakenly mixes different parts of two different tense forms: have/has which is used to construct the Present Perfect (together with V3 forms), and the V+ing form which is used in the Progressive tense (with the verb to be).

D - While this answer choice eliminates the Subject Verb Agreement mistake from the original question by changing the plural verb have into the non-conjugated V+ing form having, this change creates a new grammatical mistake, by turning the sentence into a fragment.
Think of the semicolon ( ; ) as separating two full sentences. Each sentence needs a conjugated verb.
E - Correct! This answer choice corrects the Subject Verb Agreement mistake in the original sentence by changing the plural verb have to the singular verb has.
In addition this answer choice corrects the ungrammatical use of the possessive in the original sentence by changing the phrase safety's importance to the importance of safety.