What methods are there for continuously monitoring the current in a 3-phase servo motor for a CNC machine and giving an alarm when the current is less than some value?
You need a current transformer (or three).
If you connect a resistor to the transformer's secondary winding a current through the sensed wire will cause a voltage across the resistor. The voltage is proportional to the current so will be a sine wave.
If you'd rather like a DC output proportional to the current you can use an i-Snail-V current transformer. It's self-powered and the i-Snail-V10 version for instance gives you 5 V out for 10 A. Other versions with other sensitivities exist.
A variety of current sense techniques exist.
Hall effect sensors are nice for measuring large currents through a wire at high voltages. Some have analog out, eating up one of the analog inputs on your microcontroller. Others have an integrated internal ADC, with digital pins that directly connect to your microcontroller. A few also have an integrated power FET driver, and are smart enough to unconditionally turn off the FET when it measures over-current. Perhaps you could somehow use that output to drive an alarm instead of a FET.
The Allegro Hall effect sensor chips look nice.
Related: How can I sense the motor's current?