A case for the existence of human races is best made by answering two questions:
(1) is there sufficient variation within humans for there to be subspecies?
(2) if so, is there sufficient variation to sort data into discrete subspecies?
Let's address (1):
When comparing humans to other species with sufficient variation to necessitate subspecies (race), we find that humans reach an comparable levels. Woodley (2009) compared the heterozygosity in humans with other species, all of which had wide ranges (https://lesacreduprintemps19.files.wordpress.com/2011/06/woodley-2009-is-homo-sapiens-polytypic-human-taxonomic-diversity-and-its-implications.pdf). Just so we're clear, heterozygostiy is the probability that, at any given gene location, two organisms of that species will have a different alleles (gene variant) at that specific location. Despite humans being anywhere from 99.5-99.9% the same (humans are 98.7% the same as chimpanzees, too (http://humanorigins.si.edu/evidence/genetics)), it is possible to have more instances of variance than similarity, it's just that the magnitude of those differences accumulates to 99.5-99.9%.
Woodley's data is under the heading "Table 2" in the first link provided.
His conclusions (found under the heading "discussion" were that,
"There are strong grounds for suggesting that the hypothesis thatH. sapiens is polytypic rather than monotypic is at least plausible ... Firstly, it has been demonstrated that there exists a considerabledegree of diversity (as measured by morphology, heterozygosityand FST) within this taxon, which is structured in such a way thatis suggestive of the existence of around five major clades (continentalpopulations) corresponding to biological subspecies. Andsecondly, as the phylogenetic species concept does not recognizethe validity of subspecies as a division, opting instead to labelthe most basic monophyletic unit as ‘species’, a case could be madefor the minor clades (sub-continental/racial populations) withinHomo qualifying as phylogenetic species in their own right, especiallywhen considered in light of the evidence suggestive of theidea that lineage admixture is in fact fairly peripheral and is probablynot negating the evolutionary distinctiveness of those groups."
Thus, yes, there is sufficient variation between humans to warrant subspecies.
Now let's address (2):
The best way to sort human genetic data into discrete groups (subspecies) is through correspondence between genetic cluster and geography. Using as little as 3 human subspecies categories (K=3: African, Asian and European), Bamshad (2003) was able find almost 100% correspondence between cluster genetic and geographical location, given that 160 loci (a fixed position on a chromosome) were used (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1180234/).
Alloco (2007) conducted a somewhat similar study, looking at random locations of SNPs (a variation in a single nucleotide that occurs at a specific position in the genome). Using only 100 randomly selected SNPs, 97% correspondence between self-reported ancestry and best-fit genetic cluster (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1828730/pdf/1471-2164-8-68.pdf).
In both cases, there was sufficient variation to sort data into discrete subspecies (race), as pre-defined races fit genetic clusters with near-perfect accuracy.
Hence, the human races exist.