More like circumstantial evidence I suppose, but after some folks had noticed a whole series of systematic relationships between languages in Europe and South Asia, they started to compare these languages together mainly phonologically and semantically, deriving a bunch of rules of language change which when applied to the daughter languages led to the reconstruction of the hypothetical proto-language, Proto-Indo-European, from which they had descended. Not all folks believe in the genetic relationship between these daughter languages, appealing to other linguistic processes (called areal). The Russian prince, NS Trubetskoy wrote an engimatic article shortly before he died (as a result of an intense interrogation in Vienna by the Gestapo), but some think it was a joke (he had delivered it as a talk). Others, e.g. RMW Dixon, have pointed out that these areal processes fit some languages better, e.g., Australian Aborginal ones.
I personally hold that the IE languages are related genetically, i.e., that they are descended from a common ancestor, and that we have some idea of what that language's lexical element looked like. The comparative method works rather well with the Romance languages, which few would argue are not descended from Latin or something a lot like Latin.
