The easiest (and maybe cheapest) way is to book a transfer through your cruiseline. The other options are motorcoach, railway, or private transfer. The first two require a trip into London to catch a connection. A private transfer can get expensive.
There are four different ways to travel from London to Dover. Dover is approx. 100 miles from London.
1. National Rail The bad part is you would have to take a taxi to the station from where it leaves from. There are probably about 20 - 25 stations that you could leave from. a anyway it takes about 2 hours and costs approx. $33.
2. National Express Bus takes approx 4 - 5 hours and costs approx. $19. Not so good for people in a time crunch.
3. Private car. A private car costs $185 and a van costs $195. Takes about 2 hours. OK if you have a group to share the expense with.
4. Or of course your cruise shuttle which I agree with the above statements that you should take if you are arriving the same day unless you have lots of time.
You don't have to go into central London from either airport.
1)from Heathrow National Express intercity buses ("coaches") go every hour or so direct to Dover, takes around 4 hours but no change of bus (from Dover to London best to book by computer as
coaches get crowded in rush hour)
2)from Gatwick National Rail (train station in airport) have to change trains but only about 2- 2 1/2 hours.
Both work fine; have done both. National Express and National Rail both have good websites