How to interpret 尽一份责任？
尽  http://www.cantonese.sheik.co.uk/dictionary/characters/2662/
[3] [v] try one's best; devote to

尽 in 尽责任 means 'try one's best to  fulfill'
'尽一份责任' means 'try (their) best to fulfill (their) share of duty'
More examples of 尽 (v):
尽一分努力 (devote all effort one can summon )
尽人事 (exhaust one's ability)
尽心尽力 (exhaust one's wit and strength)