Correct?
1. The librarian didn't object to the reader keeping the book one day longer.
2. The librarian didn't object to the reader's keeping the book one day longer?
Are they both correct? And what grammatical structure do we use in the first sentence? Is it participle?

Question

Answer

The second sentence is technically correct, as the gerund requires the possessive form if it belongs to someone/something else. HOWEVER, we have basically forgotten this little grammatical rule, and so leave it out in everyday language.

Answer

Both are correct.  The first is more common.

This is a gerund with a specified subject.

More information here:
https://en.wikipedia.org/wiki/Gerund#"Gerund"_clauses_with_a_specified_subject

Answer

First is correct, but the 2nd form  is never used in conversation and rarely if at all in writing. 
The problem is, if you use the 2nd form in conversation, people would always misunderstand you and think you are saying readers', the plural form.