Do step children have rights to money after death of both parents?

Second marriage for both father and mother. Married for 39 years in NY. Father has three grown children and mother has six grown children. Father died May 2014. If mother passes away before she can sell her house (which was jointly owned and she inherited from her husband when he passed), how would the house and assets be distributed? If mother sells the house, do the fathers children have legal recourse to the proceeds from the house (when she passes)?

If one parent dies first and the other parent then inherits from the deceased parent. The children of the initial deceased parent no longer have any claim on their parents estate since it is now the step parent's estate.

Of course if the step parent mentions them in a will then they become a beneficiary of that will.

However, if the natural parent dies intestate, its probable that the natural children along with the spouse would have inherited some portion of the estate. The surviving spouse's natural children would not have inherited anything.

To be specific about your last sentence,
If mother sells the house, do the fathers children have legal recourse to the proceeds from the house (when she passes)?the answer is no recourse. She'd have to mention them in her will.

I'm having a hard tiome following the sequence of events. I assume "father" and "mother" were married at the time of father's death, and consequently mother inherited the house. She has not remarried since then, correct? If she were to die without a will and unmarried then her estate would be divided among her natural born children and none would go to her step-children. In other words, even though the father originally owned the house the children of the father (the mother's step-children) would not get the house upon her death (assuming that the father himself left no will - is that the case here?). It doesn't matter whether she sells the house or not - all of her estate assets get divided this way. Of course dividing ownership of a house among her six children may be problematic for them and the executor, who may ultimately decide it's best to sell and divide the proceeds, but that's a different issue.

In a case like this its important that the mother understand that if she has no will she is explicitly accepting this arrangement as what she wants. She is also accepting that the court will have to name an executor, and if the six siblings are not all of like mind on this it can get quite messy. She really ought to have a will drawn up.

... Father died May 2014 ...

That month hasn't happened yet. I assume Rocky means 2013.