In continuity check firing systems, you apply a voltage, lower than what is required to fire the detonator, to check continuity. If you can, selectively, connect the firing circuit branch high side to an Arduino Supply pin (+5, or +3.3, if you are using a 12v firing circuit); and the firing circuit branch to an Arduino Digital pin which has a 10K Ohm wired to ground (reads zero, low, when nothing else is connected), you shoule read a one, or high, to indicate coutinuity, or a zero, low, to indicate open circuit (no fuse connected, or fuse burned open). You should be at safe voltage/current levels to avoid inadvertantly setting off the fuse. The only problem with this setup is that you have to have a way to compleetly disconnect the Arduino (power and digital input) from the firing circuit, before firing (otherwise you WILL damage the arduino).
A safer and simpler way is to wire a conventional type firing circuit with a D.P.D.T. throw switch on each branch: one side of the switch connects to the low voltage test circuit (a low voltage source and an indicator, like a LED or a lamp); the other side of the switch connected to the Firing supply buss (+12V, in your case, buss connected/disconnected by another switch, which is the master arming sw); the common poles of the switch connected to the firing branch (the lines to the fuse).
1) disarm the system by opening the Master Arming Switch (good idea to disconnect your 12V source also).
2) wire in the firing Branches; place the fuse, connect fuse to branch, repeat for all other branches.
3) confirm circuit test switches are in center/neutral position.
4) connect branches to respective terminals on firing box.
5) connect test voltage to firing box.
6) circuit continuity test: throw each branch switch from neutral to test, confirm indicator on, return switch to neutral position. Repeat for each branch.
7) decide firing mode; all at once, or individually fired.
if firing mode is all at once: verify master arm is off/disconnect/dissarmed, throw all selected branch switches to fire position.
if firing mode is individually: verify master arm is off, verify all branches switched to neutral position.
Call firing warning and wait for all clear.
9) Verify firing supply is reconnected to firing box and master arm is off.
10) FIRE: if mode is all at once, switch master arm from off to fire/on/connected.
if mode is individually, switch master arm from off to fire, switch branch switches to fire position in desired order.
11) verify successfull detinations and follow post firing safety precautions switch all switches on firing box to off position, disconnect firing supply, disconnect all branch circuits and shunt branch lines (short them together). call all safe and go check the damage.