On the last page of chapter 18 of Ireland and Rosen's book, they say that Weil defined zeta-functions of smooth varieties over number fields in his 1954 paper "Abstract versus Classical Algebraic Geometry". It's in Weil's Collected Works, vol. II, pp. 550--558.

Edit: In an article by Lang about the history of the Taniyama--Shimura conjecture, available at http://www.ams.org/notices/199511/forum.pdf, he attributes the conception of the zeta-function of a variety over a number field as a product of local factors to Hasse in the 1930s. (It was too early then to get the correct definition of local factors at the bad places and Hasse's conjecture was a *meromorphic* continuation and functional equation.) Weil is credited with bringing the conjecture to a wider audience at the 1950 ICM. I don't know if Hasse in the 1930s was really thinking about varieties of dimension > 1 or just curves. After all, number theory in the 1930s was still pretty one-dimensional when it came to the overlap with algebraic geometry. Since Victor's question is about the case of curves, then the answer to the question appears to be Hasse.