I have several simple table radios, all transistor, made between the
70's and 80's, and 2 or 3 of them hum, even when they are turned off.
The humming is coming out of the speaker. How can that be?

In each case, the on/off switch must be working, because when I turn it
on, a radio program comes out. When I t turn it off, only the hum. In
one case, I put a separate switch on the wire to the speaker, and now
the radio makes no noise when it's off.

One possibility is that these radios have some sort of
interference-suppression or anti-kickback ("anti-pop") capacitor
bridged across the on/off switch. This capacitor would allow a small
amount of current to flow "around" the switch contacts when the switch
is "off". If the cap is becoming "leaky" with age there might be
enough current flowing to partially power up the radio even with the
switch in the "off" position.

Another possibility is that it's mechanical hum rather than
electrical. If the on/off switch is in the secondary (low-voltage)
part of the power supply circuit, then there would be mains voltage
present on the primary transformer windings at all times. The
transformer could then be humming a bit, from magnetostriction, and
the speaker cone could convert this mechanical vibration to audible
noise. [This wouldn't account for your saying that adding a switch in
the speaker wiring got rid of the hum, though.]

Do these radios have any sort of clock or light or ??? which would
need power at all times, even when the radio is turned off?

Model number(s)?