People with a strong *sense of* responsibility can always gain other people's respect and admiration, and furthermore, their relationships will benefit from it. We wouldn't say "strong responsibility". It sounds similar to "heavy responsibilities," which means that your job is important and there are major consequences if you fail. A strong *sense of responsibility* means that you always meet your obligations "Bob is lazy and unreliable, but his father gave him a job with heavy responsibilities. He is in charge of a nuclear plant. He needs to get a better sense of responsibility soon. If he doesn't, people will die."

I guess that you want to say something like It is desirable that people with a big responsibility earn the respect and credit from others; that would be beneficial for their relationship.

I don`t understand the relation between having a strong responsability and winning the respecto and credit from others. Why the first could always make the second happen?