All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
According to Sharia, step-daughter born of wife with whom marriage has been consummated is called "Ar-Rabibah" and she is amongst the women who are permanently forbidden to step-father for marriage. This is attested to in the following verse:" Prohibited to you (For marriage) are:- Your mothers, daughters, sisters; father’s sisters, Mother’s sisters; brother’s daughters, sister’s daughters; foster-mothers (Who gave you suck), foster-sisters; your wives’ mothers; your step-daughters under your guardianship, born of your wives to whom ye have gone in,- no prohibition if ye have not gone in."{An-Nisa`, 23}.
Based on this confirmed permanent prohibition for marriage, it is allowed for the step-father to stay in private with step-daughter, look at her, and shake hands with her just as he does with the rest of his mahram women (Women to whom marriage is forbidden). Moreover, it is allowed for a man to reside with his mother, daughter, paternal and maternal aunt, step-daughter and the like. However, this is provided that the teachings of Sharia in this regard are observed in terms of their dressing modestly and what is allowed to be looked at from these mahram women (Hands, hair, feet) while at home doing their daily house work, in addition to making sure that there is no temptation.
It is stated in the Shafi`i book "Fateh Al-Khareeb Al-Mojeeb,P.231" :"According to textual evidence, there are four women forbidden for a man in marriage: wife`s mother (Real mother or foster mother),whether marriage has been consummated with wife or not, and step-daughter….. and all of these are permanently prohibited for marriage."
In conclusion, there is no sin when step-father resides with step-daughter since she is considered a mahram (Individual to whom marriage is impermissible) of his provided that there is no temptation and modesty is observed as regards the code of dress and what he is allowed to be looked at as mentioned above. And Allah the Almighty knows best.