The only offences which will prevent you from legally holding an ASIC card are known as schedule 3 offences, and are those which include dishonesty, violence, drugs (maybe some others). This is the same across the country, so for more detail check with your local airport authority as to whether they would view your offence as a schedule 3 offence.
Although your offence may not be a schedule 3 offence, it may still however appear on your AFP Criminal History check as conducted by the airlines, and whether this is a determining factor in your potential employment............
Driving offences generally don't appear if they are expiated, however more serious driving offences where a conviction is recorded will appear.