As far as I know, only two of the wires on the alternator really matter. The obvious big one and the small one that goes to the ignition switch (might be labeled IG terminal). There should be battery voltage at that small wire with the ignition on.
Checking continuity on the big wire might not be enough. A corroded connection (maybe at the fuse?) might be able to pass enough current for a meter, but not for charging the battery. Try measuring voltage between the alternator main terminal and the positive battery terminal with it running. Voltage should be 0 or very close to it. Same with the negative battery terminal to the alternator housing. After typing that, I see 2000mc already suggested it.
