Protecting laser diode that is transmitting an audio signal
I want to operate the laser at the highest power without burning it
  out, so how do I do this?


You are probably over driving the laser diode:


  An often overlooked factor in handling diode lasers is the influence
  of temperature on the relationship between optical output power and
  operating current. While the threshold current raises with temperature
  the optical output power and differential efficiency decrease. The
  driver circuit thus should have a safety feature that ensures that a
  significant temperature increase will not destroy the laser.
  
  Any driver circuit for diode lasers should include a well-filtered
  power supply that, as efficiently as possible, blocks inductive loads
  and other sources of interference. Battery operation circumvents the
  problem but is not an option in many industrial applications. Keeping
  the connections between the diode laser and the driver circuit short
  generally will help reduce interference.
  
  Integrated driver circuits offer a variety of functions and safety
  measures, and they require few additional components. A simple
  resistor trimmer can usually set the operating point. It is important
  that diode lasers always have a regulated driver in either automatic
  current control or automatic power control operation. A standard
  laboratory power supply is not suitable for driving them directly.


Source: Driving Diode Lasers

So make sure you have clean power, get a good diode laser driver circuit and cool down your laser diode if necessary. 


  And when they say use a 'constant current source' what does that actually mean? When I look them up, I either find really large power
  sources where the lowest they go is 1A (I need 14mA). Or I find these
  small components that clearly need something else to them (like the
  LM317). What would you suggest for my problem?


See also this question: why-is-a-laser-supply-so-tricky

A constant current driver could be as something as simple as a transistor current mirror or with an opamp regulating it. In your case you would want to make sure that the CC circuit would not deliver too much current to the diode. 



Here is more info on some good circuits for constant current circuits.


  Also, because the laser is transmitting the audio signal, the voltage fluctuates ever so slightly to create the signal- will this affect the
  solution?


Yes, if your modulating the current it will change the brightness of the laser. Since a laser diode has a linear voltage-current curve you'll need to change the current up and down with a constant current circuit with an input that you can control.  If you change the + terminal out for a voltage you can control in the circuit shown above with a voltage source then you can modulate the current. Again, with the circuit above you need to calculate the current output and make sure the voltage input stays below a value that will not exceed the rating for the current of your diode. The equations are listed on the circuit.