How likely is it that brother-sister incest will produce a healthy child?
Taking the question exactly as it is written, a very small percentage of sexual encounters result in a live birth. Only sexual encounters that result in a healthy sperm entering a healthy egg and fertilizing it, followed by successful implantation in the uterus, followed by gestation until viability will produce a healthy child. (This is leaving aside reproductive technologies, donors, and surrogates.)
But I suspect that the question the person was really asking is that if a brother and sister have a pregnancy that comes to term and is successfully delivered, what are the chances of the child being healthy?
Most likely, with any given brother and sister, the child will be healthy. With any birth, there is a chance of the child having a birth defect or genetic problem. All it takes is for a genetic problem of one or both parents being passed along, or a the embryo developing one on her own. If two people have the same genetic problem, as siblings may, there is an increase in the chance that their child will have the same problem in comparison to the general population. However, that increase in chance will still mean that the overall chance is a minority chance.
Anyone concerned about these things should have genetic testing and counseling. People who are not close relatives can also pass along health problems.
We do not deny people their freedom to marry or their sexual or reproductive rights based on their lack of ability to birth healthy children or lack of intention to birth children. We do not make criminals out of them based on those things, either.
There are healthy, intelligent, attractive people whose birth parents are genetic siblings. We all know some, whether we know their biological parentage or not, and whether they know their true biological parentage or not.