Base resistor on transistor
You have a bit under 20mA going to the LEDs (the Vf's are not all that well controlled, so let's use 20mA). 

For the 2N3904 the Vce(sat) is guaranteed at 10mA and 50mA (bracketing your 20mA collector current) at Ic/Ib = 10. We can relax that to 20 without any serious problems since hFE is between 60 and 100 minimum (measured with Vce = 1.0V). All this stuff you can find in the datasheet.

So you want 1-2mA base current. 

Since the Arduino nano has a 5V supply, the output will be close to 5V if you don't load it too heavily (you'll have to check the ATMega328p datasheet for the characteristics). 

We know the base voltage is about 0.7V when 'on', so the voltage across your base resistor will be 4.3V or so. So the resistor should be somewhere around 4.7K to 2K roughly.