I only know the train connections and usually work in terms of where on the Circle Line I will arrive to decide which one to use but:
Stanstead Express approximately 45 minutes and runs every 15 to 30 minutes depending on the time of week/day - leaves you in the north-eastern part of the circle line, Liverpool Street has the Circle, Hammersmith & City, Metropolitan and Central lines running through it and is one stop from Bank station. It also has good bus links with eastern and central London and the bus stops are easy to find. Stanstead is the furthest airport out from London of the three as far as I know. The train is pretty expensive and it does seem quite slow - also it doesn't provide much in the way of alternative connections (unlike say Gatwick where you have the faster or the cheaper!)
Luton: bus to the train station, train station links pretty frequently to Kings Cross Thameslink which is near to Kings Cross on the north of the Circle Line (with the Circle, Hammersmith & City, Metropolitan, Northern, Victoria and Picadilly lines running through it and pretty good bus links nearby but not so easy to find them!) and London Bridge just south east of the Circle Line (with the Northern and Jubilee lines, one stop to Bank and excellent bus links). I find this one the most onerous one to do as the train is just the normal train to Brighton and can be quite slow, plus the bus journey at the start/end and the walk to Kings Cross if that is where you get off.
Gatwick: Very good train links, both fast (Gatwick Express) and the not so fast. Train is about 25 minutes in if I recall correctly, slow trains link to the southern part of the Circle line, Cannon Street or Charing Cross or to L'Bridge - see above, Express goes to Victoria which is on the south western part of the Circle line (with Picadilly, Circle and District lines, good bus links to central London and the main intercity bus terminal)
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rail website for more details and for cost and time comparisons