My background is in the biological sciences, not in history proper. However, seeing as we are considering a question potentially linked to the historical application of antibacterial silver, I thought that I could find something useful from a different perspective.
I happened upon a 2009 publication from Surgical Infections titled History of the Medical Use of Silver. In the article, James Wesley Alexander lists a number of claims about the early use of silver. These include:
All of these claims are tied back to a single secondary source of Alexander's: Argyria: The Pharmacology of Silver by William R. Hill and Donald M. Pillsbury. Provided that this all hold true - again, I'm not a historian, but as a scientist, I'm willing to put some trust in scientific publications - one could see a potential path of logic for someone to determine silver as an anti-supernatural metal as an extension of its historical use for anti-"wrongness."
However, just because the path can be seen doesn't mean that's how things actually took place. Without historical evidence that this jump occurred, I'm afraid this is just a basis for conjecture.
Interestingly, the Alexander article does go on to suggest that the medical use of silver really took off in the 20th century, which seems to coincide with its rise in popular culture in the response by /u/AncientHistory .
The idea of magical weapons is old and widespread. The idea of silver, specifically, being the secret ingredient that makes a weapon magical and/or special is relatively rare. Stith Thompson in his Motif-Index of Folk Literature (1956) identifies Motif D 1385.4: Silver bullet protects against giants, ghosts, and witches, locating it in Japan and North Carolina folklore (one example in each). This is extremely meager showing for an index that often provides dozens of examples of a motif. In addition, we should not look either to Japan or North Carolina as the source of this motif. This is imply an odd, rare eruption in the international folk imagination, unrelated and of little significance.
In addition, we need not look to any particular properties - other than the face that silver is heavy and valuable - to imagine how the motif was conceived, either in folk literature or elsewhere.
I am not prepared to discuss the history of the motif of the silver bullet in printed literature and/or in film, but my guess is that this is where it was conceived and the reason why it has spread. The motif of the silver bullet does not appear to be a borrowing from folklore.
Now we need to someone with a background in horror literature and film to address the history of the motif in that context.
I seem to remember being told in the past that this was because Judas betrayed Jesus for silver, and was damned by this. So somehow silver would later be used to defeat creatures of evil, due to religious reasons.
Could anyone speak to that, or know some sources that explain or expand?