How would one go about bringing the mass of Jupiter under control? As with probes to the surface of Venus, the conditions prevailing in the interior of a gas giant would be extreme. On the other hand, they are much less extreme than in the interior of a star. And for a space faring civilization that is mature, it would seem like a credible goal. Surviving the extremes then becomes a quest to use in-situ materials to reproduce and evolve. Perhaps finally it is a question of mass utilization. Jupiter and the other gas giants are huge sources of raw materials for an otherwise empty solar system. Dyson sphere time.
Part of the challenge of dealing with Jupiter is that we are creatures of the solid state. That is what we are accustomed to arranging and building with. Three possibilties come to mind in exploiting Jupiter's mass.
All three paths are immensely difficult. There is also the question of whether the mass is kept where it is (inside the gravity well) or somehow liberated therefrom. We do not know what technologies we may later develop, but transmutation of hydrogen to heavier elements would be cool.
Given the need for large amounts of energy to do transmutation of elements, how does one obtain such energy? The Sun generally releases a large of amount of energy by "burning" hydrogen into helium. On a small scale it may be possible to actually produce energy during transmutation, as with standard Fusion and Fission reactions. These, however, are the easy transmutation reactions. To fuse heavier elements or to split lighter elements, it may require vast amounts of energy.
Jupiter has a pretty deep gravity well. It may end up being easiest to leave the mass inside its gravity well. On the other hand, if one wants to get at the surface area that all that mass represents, it may be necessary adjust the gravity well. In a simple case, such as breaking up an asteroid, the potential energy needed to divide the gravity well is relatively minor. To break up a planet like the Earth (a la the Death Star), a tremendous burst of energy is needed. Consider the energy required to boost a single kilogram out of the Earth's gravity well. Now multiply that by the mass of the Earth. Finally, when we think of Jupiter, we need to go up several more orders of magnitude. Not very easy, by any means.
Actually solid and liquid. We have to think for a moment who we are digesting Jupiter for. Chances are, Humanity as such will have given way to more advanced, semi-engineered beings. Whether our active elements will be Carbon or Silicon may be irrelevant, since most of the mass we care about for industrial purposes will be structural. And that can be any of a variety of materials. Primarily that which you have available. In a solar system of moons and small, rocky planets, metals and silicates seem to dominate. When the civilization has consumed and utilized all the mass available in the moons and small planets, it will be time to turn our attention toward the Gas Giants. Yet because the environment herein is so different, it may be necessary to create new beings, or at least new habitats, for which the high temperature and pressure is more normal.
Can creatures be created that will find the interior of a gas giant "comfortable"? What resources are legitimately available to them, and what extremes will they have to work with? Will it be easier to deal with a smaller, cooler Gas Giant like Neptune, versus the mammoth proposition of Jupiter? As with the creatures that life at the bottom of the Earth's ocean floor, around the geothermal vents, Gas Giant life forms would congregate around sources of free energy.
One must assume that any civilization capable of seriously considering the exploitation of Gas Giants is going to have some pretty nifty magic. The first attempt at Gas Giant utilization may be the capture of gas (primarily hydrogen) in a fly-by mission. If Bussard Ramjet Fusion scoops have been developed, a similar technology may allow small amounts of gas to liberated or captured in missions that are never per-say trapped by the gravitational field of the planet. More extreme methods, such as surrounding the planet with a small ring-world or hula-hoop, may allow for a more efficient extraction of certain substances from the interior. Upon finding a solid core, conditions at the surface might or might not allow for Chemically Composed solid structures to be landed and to survive for a finite time (consider the brief landings on the surface of Venus).
Protection of a landing craft may be made possible by active systems, such as laser cooling methods or magnetic bottles. Return from the surface may be made easier by the harnessing of anti-matter for propulsion. Or it may not be necessary, since all the data travelling down on the craft may be duplicated elsewhere, and information collected at the surface would be transmitted out. The reasons for serious landings would primarily be resource exploitation, and for this it would be necessary to develop physical systems which function under such extreme conditions.
What might we expect, after a planet like Jupiter or Neptune has been digested? If there has been little transmutation and the bulk of the matter is still in the gravity well, we would expect that the solid and liquid matter at the core would be under the control of our Ultra-High technology systems. Useful computation and work would be taking place at many, if not most, of the locals inside the gravity well. Some mass would occassionally need to enter and leave the gravity well, but primarily the planet would become a vast information economy many orders of magnitude larger than anything on our current horizon. The temperatures and pressures found inside the planet would be made compatible with the information processing life forms, or that the information life forms would made compatible with the conditions therein. As with the turbine blade, which functions at temperatures past its melting point, these life forms might have means of living beyond expected critical points.