The phrase "important responsibilities" is correct and usable in written English.
Example sentence: As a manager, I have a variety of important responsibilities to fulfill.
That is one of the Port Authority's most important responsibilities.
He has very important responsibilities to carry out there".
She's losing the chance to execute on important responsibilities.
Gooden and Prudhomme had two important responsibilities for the summer.
The head of this office will report directly to me and be charged with important responsibilities.
Government has important responsibilities for public health or public order and civil rights.
"One of the most important responsibilities a senator has is oversight of the intelligence community.
One quarter of respondents could think of more important responsibilities for the government.
Make a checklist of its five or six most important responsibilities.
This meant that the spokesman for Hailsham's considerable and important responsibilities in the Commons was Freeth.
I've learned important lessons from this experience which will guide me in my important responsibilities.
