If you live in Eastleigh and you've got a cruise booked out of Southampton, the transfer is usually the last thing you think about, right up until the morning you're trying to fit four suitcases in a car boot at 6 am. It doesn't need to be that way. A short run down to Southampton Port takes about 15 to 20 minutes from most of Eastleigh, and with a fixed price agreed in advance, it's one less thing to worry about before you've even left the house.
A Southampton cruise transfer from Eastleigh typically costs from £50 for a fixed-price 8-seater vehicle, covers all five cruise terminals, and takes roughly 15-20 minutes depending on traffic and which terminal your ship is using. Book ahead, especially in peak cruise season (May to September), and make sure your driver tracks your ship's return schedule rather than working off a fixed pick-up time.
Eastleigh sits close enough to Southampton that the transfer itself is rarely the problem. What catches people out is everything around it: not knowing which of the five terminals their ship is using this time, misjudging how long it takes to get bags out at drop-off when three ships are sailing the same morning, or assuming a taxi will just turn up on the day during the busiest weeks of summer.
None of that is really an Eastleigh-specific problem; it's a cruise-transfer problem. But it does mean that a decent Eastleigh taxi service that actually knows the port, rather than one that's just looked up directions, makes the whole morning easier.
A few reasons it's worth booking ahead rather than winging it:

This trips more people up than anything else. Southampton has five cruise terminals, and cruise lines rotate between them, so even if you've sailed from the port before, don't assume it's the same terminal this time:
Check your cruise line's paperwork before travel day. A driver who runs this route regularly will know current drop-off points and access at each one, so you're not walking the long way round with heavy cases.