Interestingly enough, I find this question strongly related to another one Powering the interplanetary trade ships of the 23rd-24th century and to different trade and space-economy questions we had recently on WB.
In my answer to the Q about trade ship, I basically have shown that even pure hydrogen is useful as the chemical reagent for extraction metals and(or) reducing oxides into the water and pure elements. In my opinion, it worth to spend 800 ton of fission fuel to be able separate elements of an 1 km diameter asteroid.
As result, it may help to build different constructions in space which may help way much better to Earth then just having some amount of 3He for energy production. Constructions in space may solve same energy problem as 3He, but they can not only satisfy our energy needs but also regulate climate on the earth and do other planetary stuff.
Hydrogen can be used as reactive mass for reactive propulsion for ships, which is important - space ecology, use an element with high abundance etc.
So, the answer to the question "interesting resource" is - Everything. Everything from a Gas giant is interesting and makes sense to haul in bulk quantities if you have a technology for that.
Composition and uses
not very scientific source for atmosphere compositions as a table.
Jupiter is the closest (except the Sun) source of molecular Hydrogen, which may be used as a reductant for many processes, and as a cheap method to bind oxygen and to store the oxygen in form of water, instead to lose it, release it, or try to keep it at cryogenic temperatures.
All gas giants are good sources of hydrogen because the closest place where one can find hydrogen free floating in form of asteroids is a distance about 900 A.U. (or probably even further, if at all hm, not sure and lazy to calculate, but there a temperature will be about 10K and hydrogen may form ice)
Hydrogen - reductant, reactive mass, cheap media to bind oxygen, cheap media to make water
D, 3He, He, Carbon, N - are just byproducts, each one has its own use, they are very useful byproducts.
The answer how to get is a bit harder than to answer what to get.
Orbital ring like this may help to solve the problem, but they them selfs have technological problems which need to be solved.
There a lot of problems have to be solved before mining resources from Gas Giants will be viable. Winds, Gravity, delta-v - not the first problems here.
The energy source is the first one to solve - because to lifting matter from Jupiter and launching it is energy expensive. Escape velocity for Jupiter is 59.5 km/s and that is 28 times more expensive energy wise than for Earth. Thermonuclear reactor with ease will solve the problem and may be the biggest consumer in the solar system for any 3He you may mine from Jupiter.
An orbital ring or good thermonuclear engine may solve the problem with delta-v. The ring will be more energy efficient, but to be built it need significant demand for resources from Jupiter or any GG.
A bit more advanced tech - build a ring, use thermonuclear reactors and thermonuclear engines, space cables - and no problems, but as for now, we (at the moment) do not have technology which may help us to harvest gas giant resources.
Moons of gas giant are more viable and interesting options - carbon, carbon-hydrogen, N2, NH3. (water in asteroids -> Ceres) And those resources may be used to build rings, reactors etc when time comes.
Basically, we need a ring from the answer, space cable, fission rocket or better, fission reactor or better - and we are ready to send the Everything from a gas giant. All supply fro light elements will be from gas giant for a looong time, 50 years, until we make that active matter (almost entire answer is relevant to extraction, but first part to look for technology is Note about Venus scrap, snake elephant). Carbon nanotubes are the material of the Future, which allow us to solve all problems with extraction matter from gas giants, in this or another way, the only thing we need is to master making longer tubes and making devices from them.