Check with your travel agent many cruise lines will fly you in to London and bus you to Dover. You could go into London the Paddington Express does Heathrow to London in 15 mins but costs around £30, but then you would need to transfer to a station to catch a train to Dover. I have never gone from this port so can't help much more, but hopefully it gives you some ideas to explore.
Fly to gatwick then you have all motorway except for last few miles folkestone to dover on a2.
We flew to Gatwick - took the trsin to Tonbridge - changed there and then another train to Dover Cruise terminal - took almost 3 hours because we waited about 45 minutes at Tonbridge. British Rain on internet can give you times. There is also a coach through National Express.
If you mean Dover, Kent, rather than Dover Barbados (which is easier!) then maybe something flies into Manston airport in Kent, otherwise Gatwick and some kind of shuttle. Give Fred Olsen a ring, they're quite helpful.
To travel to Dover Fly from Belfast or Dublin to Gatwick and then by train to Dover Priory. Book the train on-line for any train that day and the cost is about £27.00 return. www.thetrainline.com Trains leave from the immediate vicinity of Gatwick and on arrival is a short £5.00 taxi ride to the ship. TF Belfas.
Perhaps look for a flight to London then take train to Dover or flight to Southampton then train/bus to Dover.
hope this helps.