Idea #1 (doesn't work)
The Jupiter can't be a star, because it requires at least around 70 Jupiter masses, hidrogen to burn. Below that, there is not enough pressure and temperature in the stellar core to start fusion.
Idea #2 (doesn't work)
But, there is always a "nearest" thing, a "most reasonable" thing what seems to most realistic.
The first idea would be to use some different isotope. The best fusable isotope is Deuterium. On some estimations, a star purely from deuterium could be as small as only 13 Jupiter masses.
Unfortunately,
- it is still too high
- Jupiter contains only a negligible Deuterium
- which is a quite rare thing in the Universe, even because it fuses so easily. The few deuterium produced by different stellar processes, fuses quickly further to Helium.
Thus, also this idea fails.
So, the simple "just ignite it" solution from Clarke doesn't work.
Idea #3 (maybe... once)

What could work: most of the Jupiter mass is still hidrogen, which is fusable. Not easily, but it can. The hardest thing in the fusion is to "ignite" hidrogen into deuterium. Deuterium fuses (in multiple steps) to Helium4 already much quicker.
The current fusion projects try to ignite deuterium and tritium, even this seems very hard, but possible in industrial size. Likely it will be possible also with pure deuterium, although it would require probably larger reactors (to make it effective).
After that, there will be yet another, very big step needed: the way to fuse hidrogen. Just as comparison, in the Sun, a protons wait around a billion years to ignite to deuterium. After that, the deuterium nuclei fuse in seconds to helium.
On the current technology, fusing hidrogen would require probably fusion reactors not in the size of ten meters, rather in km size. Alternatively, some accelerator-driven solution could work.
Thus, the proposal is:
- Building very large, km-sized baloon cities to fly in the upper athmosphere of the Jupiter and contain the km-sized reactors.
- Build from them a lot.
- They should heat the Jupiter athmosphere to a reasonable temperature (for example, 500C would be already enough to give heat to te Jupiter moons).
Of course, no humanity will ever build so many machines. Likely their mass would be many million times of the current, yearly steel production of the Earth.
But, our current industry doesn't use very heavily robots. Maybe these reactors could be once built by large, automatized robotic factories on the moons of the Jupiter.
Also the robotic factories could be built by robots.
Idea #4 (it could work, but it is a very different solution)
Large (planetary size) solar sail could be used to reflect the heat of the Sun to the Jupiter, or more easily, to the Moons of the Jupiter. This image does it with our Moon, but the idea is the same also for the Jupiter.

(this image shows a Sun-Earth solar sail, but the same could be done also for the Jupiter)
The sails should be in the L2 Lagrange point of the Sun-Jupiter system, which is unstable, but maybe in the case of such a big size, a stable configuration can be found.

(this image shows a Sun-Earth solar sail, but the same could be done also for the Jupiter)
Although the sail should be in the Sun-Jupiter system, it could be focused to any of the Jupiter-Moons.
Although it should be very big, it shouldn't be very thick. A micron thick aluminium is already enough to reflect sunlight. Thus, its mass shouldn't be very big, what makes the idea the most realistic one, although it is very far from Clarke's original idea.