Wikipedia article on Baire category theorem and several other sources mention this paper: Blair, Charles E. The Baire category theorem implies the principle of dependent choices. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 10, 933–934. MR0469765, Zbl 0377.04011. However, I did not succeed in finding this paper online.
Still you can find this result for example in:
- Horst Herrlich: Axiom of Choice as Theorem 4.106
- Theo Bühler, Dietmar A. Salamon: Functional Analysis as Exercise 1.7.14
- John C. Oxtoby: Measure and Category, notes to Chapter 9, page 95
- Following the suggestion in Asaf Karagila's answer, I have looked also in the book Eric Schechter: Handbook of Analysis and Its Foundations. It contains this result in paragraph 20.16. (The result is stated for pseudometric spaces, but in the direction BCT $\Rightarrow$ DC, a metric spaces is used.)
- Some versions of Baire Category theorem are listed (together with references) Paul Howard, Jean E. Rubin: Consequences of the Axiom of Choice under Form 43.
EDIT: Some sources mention that the result was proved again later in this paper: Goldblatt, Robert, On the role of the Baire category theorem and dependent choice in the foundations of logic, J. Symb. Log. 50, 412-422 (1985). ZBL0567.03023, MR793122.
(G. H. Moore uses this in the introduction of his book Zermelo's Axiom of Choice: Its Origins, Development, and Influence as an illustration that some kind of database with consequences of AC might be useful to decrease likelihood of reproving already known results.)
I will freely admit that I found some of the above results simply by trying to search for some suitable phrases in Google Books and Google Scholar. Maybe you might find some further references in this way. It might be also worth checking the work which cite Blair's paper; you can try them, e.g., in Google Scholar, Zentralblatt or MathSciNet. (I do not have access to the latter.) Or simply check the books which mention this paper.