“other’s lives” vs. “others’ lives”
Solution 1:
Since you are talking about multiple others ("...the lives of others"), then the apostrophe comes after the s.
having an impact on others' lives
Solution 2:
If something is singular, for example, Bob. Then he is a person and we write
Bob's book.
Let us consider there are two Bobs and both of them share a single book.
The two Bobs' book
so others is plural. Consequently, the correct answer is
having an impact on others’ lives.