I’ll take the opposite approach from Malthus’ cite, and estimate it based on genetics. Really, though, to get a good answer to the question you’d need lots of good empirical data, which probably won’t happen.
Anyways, to keep my estimate simple, I’m only going to consider autosomal recessive mutations. You’ve got two copies of each gene, so if one is simply “broken”, the other can often suffice. For each recessive mutation carried by the grandparents, there is a 1/4 chance that it will be carried by both parents, and ultimately a 1/16 chance that it will become homozygous (and have some mutant phenotype) for the child in question.
We can ignore spontaneous recessive mutations in the parents, since those won’t be homozygous in the child. And I’m going to ignore spontaneous parental dominant mutations, since those are much less common.
So, what are the odds that the grandparents are carriers of recessive mutations? Pretty high, actually. In each generation, there are a hundred mutations new mutations (give or take – this recent measurement says 60, this one says 100-200). Most of those mutations will be harmless, though it’s hard to say how many. The standard WAG by geneticists is that ~5 of those new mutations will be harmful recessive alleles. And that’s for each grandparent.
In total, the child has ~10 1/16 chances to inherit something bad, which works out to be a 47% chance of genetic defect. (Which is remarkably close to the 12/21 figure cited by Malthus.)
But so far I’ve neglected to consider what different sorts of mutations might do. A grandparental mutation in really fundamental biochemical machinery will kill the parent’s gametes (which have to function with only one copy of a gene). That would result in infertility for the parents. Similarly, there are lots of essential development-related genes. Break those, and the child will die early in development. If that’s before it implants in the uterus, the parents will just experience more infertility. Or, if development can proceed a little while longer, they could have a high rate of miscarriages. So, to end up with a child with a congenital defect, they have to express a mutation in a gene that’s important enough to cause problems when it’s broken, but not absolutely critical for eventual birth.
I have no way to estimate the fraction of possible mutations that are important but not lethal. It’s probably a very small fraction of those hundred new mutations per person, though, maybe just one or two on average. So, my ultimate WAG for the chance of birth defects? I’ll just bound it between 1/16 or 6% (one of the grandparents almost certainly carries something bad) and ~50% I figured above, assuming several disease alleles per grandparent.