Hi Sulross, Yes London Heathrow is by far the easiest airport for Southampton, Gatwick is also an option but will add a good 30 minutes plus to your travel time.
Unfortunately neither have great links to Southampton, from LHR you either have to travel into London Waterloo for the train back to Southampton Central, from where a 10 minute taxi ride will deliver you to the port, or take the RailExpress coach to either Reading or Clapham Junction & again onto the train Southampton Central.
Make sure you get off at Southampton CENTRAL as there is also Southampton Airport Parkway (limited flights to Europe only) which is about a 20 minute taxi transfer.
From Gatwick, there are again trains into central London and Clapham Junction.
You can get a taxi (LHR = 60 miles, LGW = 90 miles) but they can be VERY expensive if you get them at the airport, so you are better booking in advance. LHR is around £100 e/w.
Southampton Airport (SOU) does have services for Amsterdam & Dusseldorf, so if London flights are looking to be a problem, check out US flights into those destinations as well. Then of course it's a simple 10 minute train journey from SOU into Southampton.
Finally, and certainly not a first choice, Greyhound & National Express run coach services from LHR direct to Southampton, and park near the docks.
Oh, and make sure you know which cruise terminal you will be sailing from, we have 4 main terminals, they are not that far apart, but an inconvenience if you end up at the wrong one. Check the website Cruise Ship Schedule - Southampton VTS - ABP Southampton
for cruise schedules.
Any more questions, just ask :o)
Hope you have a great time,