I going to assume this is the Ford Factory anti-theft (PATS) system.
Turn the key on but do NOT crank. Does the theft light come at initial key on "lamp test"? If not, then the theft light is burned out or has been removed from the cluster.
Does the theft light go out after a 3 second "prove out"? If not, then stop and find out why.
Note, I suspect that the PATS key is not been read. Why? Because if it were, the alarm would reset.
Also, unlocking the driver's door with the key should reset the alarm. If it doesn't then suspect the door key lock switch isn't working correctly. It's a simple on/off switch mechanically driven from the lock. Suspect it has been water damaged due to age.
The alarm system also depends upon the door ajar switch. So expect problems if the door ajar switch isn't functioning correctly.