I think it all depends on aircraft type, how many crew you carry, how many passengers you carry and which country you operate from.
In Australia we have a rule, 32 (or is it 36) pax to 1 crew member, and the crew compliment has to be in line with that. Minimum crew can happen (depending on the airline) just so long as the numbers add up and exits are covered. This obviously differs with other countries and even here they are trying to push for 50pax to 1 crew member.
If someone went sick say overseas it shouldn't make too much difference but if the pax load exceeded the crew requirement, from my understanding they would have to offload pax. This can cause problems for smaller a/c types. What do you fly? A 747 should be able to handle the cutback of crew but again that depends on the airline and country you operate with/from.
Hope this helps.