Which sentence is incorrect and which one is correct? Why?
- She is responsible for answering the phone.
- She is responsible in answering the phone.
1) is idiomatic English.
2) is possible, but I would understand it with a different meaning, something like "Whenever she answers the phone, she acts responsibly". It is ambiguous (as is my paraphrase) as to whether it is the act of answering the phone which the speakers sees as responsible, or the way she behaves when she has answered the phone.
Responsible for is chargeable with being the author, cause, or occasion of something usually followed by for:
Termites were responsible for the damage.
She is responsible in answering the phone.
My interpretation of this is that she is responsible whenever she answers the phone, this associates her behavior with responsibility when she answers her phone.
In my opinion, 1 and 2 varies differently.
Both of these sentences say something meaningful, so neither is strictly incorrect. However, the first ("responsible for") is much more commonly used.
However, they mean something different:
Sentence 2 might be more naturally expressed "She is responsible when answering the phone."