You're not stating in your question if the time/era of this expedition is today (fictional science expedition?), a few years into the future (hard SF?), or far into the future (space opera SF?)
Depending on the time/era of the story your characters might have access to technology that will make this possible.
If it's far future SF, you don't need to explain how the tech works, it's like Clarke said, indistinguishable from magic. Just make sure you don't invent something that makes people indestructible, etc, or you'll remove all danger. (Though the rule is usually that if the protection is magical, so are the weapons and the treats...) And of course, since the descent into Neptune may not be dangerous, you may have to add other dangers instead. (Space monsters?!)
I once did a scene in a far future SF setting where they argued about a past event like this where one side of the argument was along the line "it's innocent fun and you need to live a little" and the counterargument was "you're an idiot and shouldn't be allowed near any gas giants at all".
I suggest you still do research on Neptune. For one to get your facts right and also because the little you've already gotten in other answers here indicates several juicy sources of conflict. Research can bring up the most interesting types of problems and pain you can throw at your characters!
Hyperrealistically, you can, of course, not go to Neptune, not with today's tech, and would you dive into it, it would be a short dive with a violent end.
So I would classify your story as SF regardless of the level of realism you wish to apply.
When it comes to SF, the requirement for realism is usually, even in hard SF, subordinate to not-yet-possible technological advances (though some authors have made predictions that were later realized), the rule of cool (especially cool gadgets, technology, and science), and similar.
For hard SF, you need a technical solution for doing this (perhaps with a pinch of techno-babble handwavium to explain away things we don't yet know how to do today). You could, for instance, get away with protecting your Neptune divers with an energy field even though such a thing could possibly have effects throughout society. That problem is usually solved by making the tech brand new, experimental, and, of course, not entirely reliable. (Only desperate idiots would get into that thing!)
If you don't give a technical explanation, I'm not sure it's hard SF, so the hard SF fans won't be happy, and if it's not in space opera time, neither will the space opera fans.
Although there are, of course, other types of SF as well, not to mention other genres where the story can be "realistic" within its genre such as a comment on Jules Verne's classics along the line of "A (steampunk) Journey into Neptune". Or perhaps a comedy where something hilarious protects them with an even more hilarious result of their dive, or even a children's story where the logic of planet diving need not apply.
The goal isn't usually to be hyperrealistic or even realistic and put that in the novel (it will be super boring), but rather to prevent readers from saying "this is unrealistic rubbish" or constructing a wobbly Ringworld.