The most common and most likely answer to this crossword clue is the 4 letter word ONUS
I've seen this crossword clue in The Times
O
N
U
S
More clues leading to the result ONUS
- Responsibility ours?
- Burden that's our responsibility
- Responsibility of working with United States
- Obligation on American
- Charge we’ll settle? As we might say, it’s ____
- Responsibility of bachelor abandoning extra benefit
- Responsibility of old student group
- Burden of working with America
- Complains unofficially about limiting responsibility
- Burden on you and me
Buddy explains!

The straight part
We know that "Responsibility" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case responsibility = onus.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does ONUS mean?
ONUS noun- A legal obligation.
- Burden of proof, onus probandi
- Stigma.
- Blame.
- Responsibility; burden.