Hi Tessie, the 2N2222 is an NPN transistor. You have drawn it correctly in your diagram. If the cathode of the LED is connected to 0 volts (ground) as shown in the diagram, and the collector of the transistor is connected to +5 volts or greater (the upwards facing green arrow head), then it should be perfectly safe to connect the base to ground. To light up the LED, try connecting the base to the collector. Connected like this, the transistor is operating as what is called an emitter-follower. This just means that the voltage on the emitter will follow the voltage on the base (less 0.6 volts).
I suggest you double-check the pin-outs on the transistor and try again (maybe with a new transistor). If anything's unclear, feel free to ask more.