Transistor 2N2222A wont accept base input from arduino
if I connect base of transistor to positive terminal three LEDs light
  up


Yes, that'll probably work despite the transistor being wired as an emitter follower.


  but when I connect base to arduino it does not work. Multimeter shows
  there is 5V on base of the transistor but no current.


Because the transistor is wired as an emitter follower - the emitter "follows" the base voltage but about 0.7 volts lower. So with 5 volts on the base there will be about 4.3 volts at the emitter and each LED will share that voltage because they are in series.

That's approximately 1.43 volts across each LED and, unlesss they are infra red devices (requiring a lower forward voltage to illuminate) you won't see any glow.

The most appropriate way to drives LEDs in a situation like this is to put them in the collector like this: -



Now the BJT fully turns on to virtually a short circuit (maybe 100 mV across it) and there is almost the full 9 volts across the current limiting resistor and LEDs. It's usually the most appropriate way to simply drive motors too except i would consider using an N channel MOSFET instead of a BJT.