There's no sensible public transport from L/C.
Your ship will probably offer bus transfers to city or airport, & probably a city tour ending at the airport for those with a late flight.
Taxis at the pier are in a cartel & charge way over-the-top, eg around 4500 baht to Bangkok.
For private transfer to either airport or city, check out pre-booked transfers over the 'net, a car for up to 3 people should be around 2500 baht to the city (90 minutes plus) or rather more to the airport.
If you have a late flight, worth checking out a car for the day, taking you on a citytour of Bangkok (or for something different, the Summer Palace at Bang-pa In plus the ancent capital of Ayutthaya) & finishing at the airport. More flexible than ship's offerings & probably no more expensive.
If you want to consider overniting, book a hotel in the city (reasonably near the river) rather than the airport. In this case, fix up in advance just a simple transfer L/C to hotel - the city is easy to explore by foot, ferry, skytrain, taxi, tuk-tuk. For onward transport to the airport, cheapest taxi option is to get your hotel to book one for you - cheaper & more flexible than pre-booking over the 'net.
Give yourself a long day in Bangkok, preferably two, the city is well worth it.