Here is my general summary, you decide what is best. There are other secondary advantages/disadvantages, particlularly of Airfields, but these are the major points for combat I can think of at this time:
Airbases
Advantages:
- Cheap (4 food/2 shields)
- Defensible (units defend one at a time, like fortress... but no defensive bonus)
- Unsinkable
- No land unit movement points in/out (NOT a RR, but similar)
- Capacity = More than 8
- Build fast (one turn, with 2 engineers)
- Extends range of Air units by cutting distance
- Free upkeep
Disadvantages:
- Can get in way (accidental landings, etc.)
- Air units can be attacked by ground units
- Immobile (but can be easily removed)
- Cannot repair Air units
Carriers
Advantages:
- Mobile (but limited to contigious water)
- Air units can land "in water" (on carrier)
- Can use its speed to help recover an air unit short on range
Disadvantages:
- High priority AI target
- Expensive
- Long construction time (or more expense)
- Positioning time
- Low capacity (8)
- Poor defense against AI Air * AI Battleships
- Needs at least one AEGIS Cruiser (expensive/unhappy)
- Can cause unhappiness in home city (Democracy/Rep)
- Costs 1 shield per turn (in some situations)