You should have power all the time to the injector & the computer grounds the injector to fire. Make sure that the ground is not shorted to ground. Use a multimeter to check. Turn the key on & make sure you have power, set multimeter to ohms & put one lead to a good ground & the other lead to the connector spade that does not have power, key off, does it have any ohms readings? If it does there is a wire shorted out. If not, then turn the key on, it should show a reading & then go out. If it stays on, it could be the computer (ECU) or a wire from the computer that is getting a ground from the computer with key on. Good luck. A noid light also helps if you can get one.
Toby