Are there any scriptures that would suggest how to treat stepchildren in a second marriage? Are we as the stepparent just as responsible for the welfare of a minor child?
The Bible says little about stepchildren. Most of what it says is about half siblings, brothers or sisters that share one parent. The only thing that it specifically says about stepchildren is that a man may not have sex with a stepdaughter for that would be incest, even though she is not of his own blood. The passages are Leviticus 18:17, Leviticus 20:17, and Ezekiel 22:11.
Beyond that, it seems to be understood that a parent's responsibilities are the same whether a natural parent, adoptive parent, or stepparent. No distinction is made, anyway, between the three. When you consider what God has done for us, saving us from our sins, and that the scriptures say those who are saved are adopted by God (Romans 8; Galatians 4; Ephesians 1:5), then it is clear that God treats the adopted children (and one would presume stepchildren if it were possible) the same as the natural son. If God treats us that way, then we ought to treat children who are not our birth-children the same as if they were.