I think we need to see a schematic. If you have a proper current limiting resistor in series with the transistor base, you won't damage the Arduino.
Transistors are basically current-amplifying devices. The base-emitter voltage is around 0.5 to 0.7V when the transistor is switched on. It sounds like you've built a "emitter follower" circuit, where the emitter voltage "follows" the base voltage (ignoring the base-emitter voltage drop).
I am trying to send 12V to the relay to activate it...
... I just so happen to have 5V relays so the 12V/5V difference isn't a problem for me.
With regular (electromechanical) relays, you should NEVER run 12V into a 5V relay coil! Power increases with the square of the voltage, so with 12V you are dissipating more than 5 times the specified power in the relay coil. It's going to overheat and it might burn up!