This fact is stated in a lot of web pages, and a proof can be found in the Mathematics Stack Exchange question Why is $e$ not a Liouville number?, but I have yet to find any nontrivial historical information about this fact.
I suspect this can be found in Édmond Maillet’s 1906 book Introduction à la Théorie des Nombres Transcendants et des Propriétés Arithmétiques des Fonctions (where the phrase “Liouville number” was first coined) or in Oskar Perron’s 1913 book Die Lehre von den Kettenbrüchen, but my very poor French and my non-existent German stands in the way.
I began wondering about this when I came across some remarks on this topic in the following 1916 Master of Arts thesis:
Jewell Constance Hughes [after 1935: Jewell Hughes Bushey] (1896-1989), Transcendentalism of Curves and of Numbers, Master of Arts thesis (under Wilhelmus David Allen Westfall, 1879-1951), University of Missouri, 1916, 24 pages.
(first paragraph on p. 17) As we have already explained, in 1844 ["1873" intended] Hermite first showed that $e$ is not algebraic by proving that it cannot satisfy an algebraic equation of degree $m.$ Is it not possible to prove the same thing by showing that $e$ cannot satisfy the Liouville inequality? The following results show the efforts of the writer attacking the problem. This Chapter may be regarded as a preliminary report on an investigation of rather far reaching character which the writer hopes to carry further after this year.
(last paragraph on p. 24) As we said in the beginning, this Chapter is a preliminary report of an investigation which the writer hopes to carry on later. We have not been able to prove that $e$ violates the Liouville inequality. Perhaps it does do so, perhaps it does not. Perhaps $e$ is one of an uncountable class of numbers which fail to violate Liouville's Theorem. At the present time we do not know.