The “generational curse” that you are referring to is not a curse in the strictest sense of the word.  We are not held responsible for the sins of others, each person is responsible for their own lives.

When the Bible talks about the sins of the father being passed on through generations, it is talking about a patriarch of a family not following after God and setting a good example.  In this instance, it is more than likely his children and grandchildren and even his great-grandchildren will not follow after God either.  This is because they did not have the example of a Godly head of family to follow.  This compares to a Godly parent who can influence their descendants to follow after God.