I would normally be inclined to say that the door switch is stuck; but, this might have been caused by your fiddling with the dome light.
You had the door apart and disconnected and connected the connectors (presumably to the latch?). When you had the connectors apart did you not check the operation of the door switches as described in the service manual? You can get a copy of the SM by going to the NSX Prime Library in the red banner and following the links. If the door is still apart, get the service manual and go to page 23-198 and it shows you how to test the switch. Confirming the switch is or is not working could eliminate the need to read the rest of this post.
The safety indicator and dome light and the security system operate off the same door switch. This is the dome light / safety indicator circuit
The first thing is can you lock and set the security system with the driver's door closed? If you can then the actual switch is functioning because if the door is open (or the switch indicates open) it blocks the security system from setting. However; if you can't that does not absolutely mean a switch problem because a short circuit to ground anywhere on that wiring path can also mimic the door open.
The second thing I would do is to switch the dome light to off and see if this resolves the problem with the safety indicator. It should not because the security indicator is isolated from the dome light by diodes so the operation of the dome light normally does not affect the security indicator. If it fixes the problem, when you popped the fuses you might have damaged those diodes.
The easiest way to check the operation of the door switch and all the wiring to the door switch if the door is no longer apart is to remove the dome light to access the electrical connector. This is the pin out for the dome light connector
Terminals F and G are not connected. Terminal D is +12v and terminal E is the connection to the door switch. With the dome light removed, connect a continuity tester between body ground and the matching E terminal on the body wiring harness. Open and close the driver's door (passenger door closed). If everything is normal then the continuity should switch from closed circuit with door open to open circuit with door closed.
If it is stuck in the closed circuit position you have a wiring short to ground (low probability) or you have a problem with the switch on the latch. It is possible that when you were replacing the bulbs in the dome light if the dome light switch was not off, and the driver's door was open and you managed to short across the terminals of the bulb you may have applied +12v directly to the door switch welding its contacts closed.
If it works correctly then you have some kind of problem with the safety indicator. I don't know how you would do this when replacing the bulbs in the dome light; but, as above if you managed to apply +12v directly to those isolating diodes they may have been damaged before the fuse popped. Normally diodes go open circuit when damaged; but, they might have shorted.
So, first determine whether the door switch is working, either by direct test at the latch connectors or at the dome light connector or just trying to lock and un lock the car and see if the security system sets; but, there are other things that may fool the security system so checking the door switch circuit is the best option. If the door switch does not work then the latch has to be replaced. You definitely need the service manual for this. Poke yourself in the eye with a sharp stick which will make this exercise seem less painful. If the door switch is working correctly pull the dome light and test it as per the service manual. I don't see what you could have done to the dome light that would cause this problem; but, rule it out because it is easy to do. That leaves you with the safety indicator. Start with a function test as per the service manual. At the point you need to remove the cluster you again need the service manual and that sharp stick to poke yourself in the eye with.