You might (might) get away with a new battery or battery reset. Its also worth checking the aerial which is around the ignition barrel.
Mine had this problem about 3yrs ago and theres lots of temporary fixes/work arounds but ultimately sometimes the transponders/CODE box do just fail.
The main problem comes because the CODE box is paired to the ECU and cannot be 'un-paired / changed' - at least this was the situation 3yrs ago. In the end I needed a new CODE box, new aerial and new paired ECU. Total cost was about 700quid from memory but the bigger issue for me at the time was the wait - 8 weeks lead time from Alfa.
Trying to get a complete paired ECU/CODE box and key set/barrels from another phase I 2.0 car could also be a goer. Cant see a scrappy charging much more than 100-200 quid for them, and then just do a straight swap???
-1998 916 Spider 2.0TS Lusso-
H&S Corse stage 1.
CATCAMs stage 1 cams with Savali custom ECU remap.
305mm discs with 166 4-pot callipers.
Ragazzon exhaust mid section.
Biscione Racing 2x70mm rear section.
17" Zender Siena wheels with 3mm rear spacers.
etc etc etc!