I think you need a connection from Arduino GND to 12V ground, there has to be a return path for the signal.
You could put a 1N4148 diode between the Arduino output pin and the relay signal pin, (cathode (end with stripe)) toward relay, that would prevent 12V from feeding back to Arduino if module circuitry failed.
Is the relay signal active high, or active low. In other words does the really turn on when the Arduino Outputs 5V to it, or does it turn on when the Arduino Outputs 0V. Most relay boards are active low, in that case the diode will not work. Besides there are two diodes that are there already. One for the optocoupler, And one LED. If your signal is active high, or if it is bidirectional set it up for active high, and there will be no chance of 12V getting into it. otherwise, the diodes and current limit resistor will prevent high current from getting to the Arduino. The clamping diodes within the Arduino should reduce the voltage to a safe level.