Where in the heck did you find a 16 Ohm amplifier???
What's the application, and what's the wattage rating on the amp?
Could I just connect a 12 ohm resistor in series and deal with the reduced sound, or is there a better way?
1/4th of the voltage is 1/16th of the power and a 12dB loss.
Four 4-Ohm speakers in series (or two 8-Ohm speakers) might be a "better" solution depending on your design requirements. (Or get a different amplifier. 
)
It will still work, just don't turn the volume so high.
There is a risk of "frying" the amp. You can take the risk if a dead amp is not a big issue for you.