Is better healthcare a bane to the long-term survival of the human race?
Gene frequencies (frequencies of each allele at a given loci) in populations are affected by many things divided in to systematic and dispersive processes. Systematic process (migration, mutation, selection) affect gene frequencies in an often quite predictable manner and strength. Dispersive process (Random drift, differentiation between sub-populations, uniformity within sub-populations, increased homozygosity) are random in their direction and only predictable in their amount. Your question focusses on the systematic process called selection.

In a large population (if we ignore dispersive processes for the time being) the gene frequencies will reach an equilibrium called Hardy-Weinberg equilibrium. This means, with constant rates of the systematic processes, each locus will reach a point of stable equilibrium. Mutation and migration bringing more variation, and selection reduces it (over simplified - sometimes selection maintains variation, see my answer about why variation exists here). The frequency of each allele at equilibrium will be where these effects balance out. If selection becomes weaker against one allele it will become more frequent in the population.

By using medicine etc. to cure diseases we reduce the selection acting on a deleterious mutation thereby putting the gene frequencies in to a state of flux during which they move to new equilibrium points. So in short the answer is yes, modern medicine is decreasing our genetic fitness by allowing deleterious mutations to become more common.

I raise the question, will we ever be able to cure all genetically caused disease? In my reckoning, no, at least certainly not in our lifetimes.

It only represents a real risk if we suddenly are unable to cure diseases. This would increase selection against that allele and the gene frequencies would move back towards the equilibrium point which existed before the selection was reduced. What this means for the population is the loss (death) of those with the low fitness genotype.

The best reading you can do on this is the opening chapters of Falconer and Mackay's Introduction to Quantitative Genetics. Also read about Hardy-Weinberg equations.

HTH