As @suchiuomizu noted in the comments, the ring was not the work of men. In the same chapter of the Silmarillion in which Beren used the ring to establish his bona fides with Thingol (Ch. 19, Of Beren and Luthien), Tolkien tells us that the jewels in the ring were "devised" by the Noldor in Valinor:

His words were proud, and all eyes looked upon the ring; for he held it now aloft, and the green jewels gleamed there that the Noldor had devised in Valinor. For this ring was like to twin serpents, whose eyes were emeralds, and their heads met beneath a crown of golden flowers, that the one upheld and the other devoured; that was the badge of Finarfin and his house.

We are not explicitly told that the ring itself was crafted in Valinor, yet it seems likely. Another possibility is that the jewels mentioned above were among the treasures Finrod had brought from Valinor, and that they were crafted into a ring by the dwarves of the Blue Mountains who had helped build Nargothrond. Tolkien does not imply that the dwarves did so, but in the Silmarillion (Ch. 13, Of the Return of the Noldor) he describes the crafting of the Nauglamír:

[T]hat stronghold was called Nargothrond. In that labour [of building the stronghold] Finrod was aided by the Dwarves of the Blue Mountains; and they were rewarded well, for Finrod had brought more treasures out of Tirion than any other of the princes of the Noldor. And in that time was made for him the Nauglamír, the Necklace of the Dwarves, most renowned of their works in the Elder Days.

So, it's likely the ring was crafted in Valinor, but it's also possible it was crafted by dwarves at around the same time they created the Nauglamír. Either way, the ring was not crafted by men.

As @suchiuomizu noted in the comments, the ring was not the work of men. In the same chapter of The Silmarillion in which Beren used the ring to establish his bona fides with Thingol (Ch. 19, "Of Beren and Lúthien"), Tolkien tells us that the jewels in the ring were "devised" by the Noldor in Valinor:

His words were proud, and all eyes looked upon the ring; for he held it now aloft, and the green jewels gleamed there that the Noldor had devised in Valinor. For this ring was like to twin serpents, whose eyes were emeralds, and their heads met beneath a crown of golden flowers, that the one upheld and the other devoured; that was the badge of Finarfin and his house.

We are not explicitly told that the ring itself was crafted in Valinor, yet it seems likely. Another possibility is that the jewels mentioned above were among the treasures Finrod had brought from Valinor, and that they were crafted into a ring by the dwarves of the Blue Mountains who had helped build Nargothrond. Tolkien does not imply that the dwarves did so, but in The Silmarillion (Ch. 13, "Of the Return of the Noldor") he describes the crafting of the Nauglamír:

[T]hat stronghold was called Nargothrond. In that labour [of building the stronghold] Finrod was aided by the Dwarves of the Blue Mountains; and they were rewarded well, for Finrod had brought more treasures out of Tirion than any other of the princes of the Noldor. And in that time was made for him the Nauglamír, the Necklace of the Dwarves, most renowned of their works in the Elder Days.

So, it's likely the ring was crafted in Valinor, but it's also possible it was crafted by dwarves at around the same time they created the Nauglamír. Either way, the ring was not crafted by men.