Does the baby die?

Yes, Coyotito (the baby) dies in Chapter Six.

And then Kino's brain cleared from its red concentration and he knew the sound- the keening, moaning, rising hysterical cry from the little cave in the side of the stone mountain, the cry of death.

Kino had a rifle across his arm and Juana carried her shawl like a sack over her shoulder. And in it was a small limp heavy bundle. The shawl was crusted with dried blood, and the bundle swayed a little as she walked.