Terraforming Wiki
A terraformed Jupiter

Terraforming of Jupiter is the process of building an artificial surface above the atmosphere and making a new, life supporting biosphere. The super strong artificial surface would be built at a distance from Jupiter's 'surface' where the gravity is about 9.8 m/s^2, and given an Earth-like atmosphere and temperatures. The low altitudes will be covered with vast oceans while the higher altitudes will be continents and islands. Doing this would involve massive amounts of power and resources, so this would be done in the far future. There are so many problems in terraforming Jupiter. It couldn't be terraformed into a rocky planet like Neptune, because even its solid core is 14-18 times Earth's mass, and due to its large gravity field, the threat from space debris would be high. Due to its pressure Jupiter has the biggest liquid ocean (composed of metallic hydrogen, water, methane, ammonia and other materials). Finally, there will be a artificial barrier to block asteroids from destroying it to turn it into energy for the planet.

A composite Voyager image of Jupiter. The Great Red Spot, a persistent anticyclonic storm, is at lower left. White atmospheric bands, termed zones, represent areas of upwelling; brown bands, called belts, represent areas of downwelling. They display high-altitude ammonia ice clouds and lower clouds of unknown composition, respectively.

From Wikipedia:

Any gas giant can be enclosed in a solid shell. At a certain distance, the gravity will be terrestrial, and energy can be obtained from drilling into the planet's surface.

See also: Jupiter Simulation

Method with current technology

Another method may be to create floating gas cities in the upper atmosphere out of lightweight material. This seems to be a viable solution that can take place in the relatively near future.

Also, an Earth-like upper atmosphere of oxygen and nitrogen and water vapour can be added.

A floating gas city