Origins
The word 'genetic' entered English around 1831, formed from Greek 'genetikós' (γενετικός, pertaining to birth or origin), itself derived from 'génesis' (γένεσις, origin, creation, coming into being). The Greek verb behind both is 'gígnomai' (γίγνομαι, to be born, to come into being), which traces back to the PIE root *ǵenh₁- (to beget, to give birth). Initially, 'genetic' was a scholarly adjective meaning 'pertaining to the origin or development of something' — one could speak of a 'genetic account' of a philosophical concept, meaning a developmental explanation of how it arose.
The biological sense crystallized after the rediscovery of Gregor Mendel's 1866 paper on inheritance in peas around 1900. William Bateson, the English biologist who championed Mendel's work, coined the noun 'genetics' in 1905 to name the new science of heredity. He reportedly chose the word for a 1906 conference, and it stuck. The adjective 'genetic' then narrowed rapidly to mean 'relating to genes and heredity,' though the older sense ('pertaining to origins') persists in linguistics, where 'genetic classification' groups languages by ancestry.
The word 'gene' itself came later still — coined in 1909 by the Danish botanist Wilhelm Johannsen. Johannsen shortened Hugo de Vries's term 'pangene' (1889), which de Vries had built from Darwin's 'pangenesis' (1868) — 'pan' (all) + 'genesis' (origin). Darwin's hypothesis was wrong in its mechanism, but it bequeathed the vocabulary. So 'genetic' (1831) predates 'gene' (1909) by nearly eight decades: the adjective existed before the noun it now seems to modify.
Latin Roots
The *ǵenh₁- root family is vast. Through Latin 'genus' (birth, race, kind), it produced 'generation,' 'generous,' 'gender,' 'genre,' 'gentle,' 'genuine,' and 'ingenious.' Through Greek 'génos' (race) and 'génesis' (origin), it produced 'gene,' 'genetics,' 'genealogy,' 'genocide,' 'eugenics,' 'hydrogen,' and 'oxygen.' Through Germanic, the same root gave 'kin,' 'kind,' and 'king.'
Modern compounds built on 'genetic' and 'gene' have proliferated enormously since the discovery of DNA's structure in 1953: 'genome' (the complete set of genes, coined 1920 by Hans Winkler from 'gene' + the '-ome' suffix from 'chromosome'), 'genotype' (the genetic constitution, as opposed to 'phenotype'), 'epigenetic' (above or beyond the genetic — heritable changes that do not alter DNA sequence), 'transgenic,' 'mutagenic,' and 'oncogene.' The Human Genome Project (1990–2003) cemented 'genetic' as one of the defining scientific adjectives of our era.
Philosophically, the old and new senses of 'genetic' are not as distant as they seem. To give a 'genetic account' of something is to explain how it came to be — which is precisely what molecular genetics does at the level of organisms. The word has always been about origins; it simply found its most powerful application in the biology of inheritance.