Connecting BeagleBone Black to an Arduino
Arduino expects a minimum of 0.6 * 5 V = 3 V at ts input to detect it as logic high. There is no harm in trying once by connecting directly from BBB to Arduino. Connect both the grounds.
If the voltage dips below 3.1 V for some reason, you can go for 3 V to 5 V voltage translator. 
It should work without translator.