No manned spacecraft has ever reached further than the moon. People have sent several much smaller space probes to Jupiter. Despite all the sci-fi movies which made it look like a dangerous and dramatic adventure to maneuver through an asteroid belt, the reality is that, even within the asteroid belt, the density of rocks that poses threat to spacecraft is very low. So there is not much chance for a spacecraft to hit an asteroid. The main difficulty for men to travel anywhere beyond the moon is still the energy cost and mechanical difficulty of such a huge rocket.

We haven't sent a person to Jupiter yet, however, we've launched multiple spacecrafts past Jupiter in a bunch of flybys on their way to Saturn and beyond. Those journeys took on average 620 days and the spacecrafts were named Pioneer 10, Pioneer 11, Voyager 1, and Voyager 2. They were launched in 1972, 1973, 1977, and also 1977, in that order.

We've also sent a much slower spacecraft to orbit Jupiter called Galileo (launched 1989), which took its time using the Earth's and Venus's orbit to slingshot it to Jupiter. Galileo took 2,242 days to get to Jupiter. The reason why it's so much slower than the other spacecrafts is because new safety rules at the time required Galileo to have a lower-powered booster.

As for going under the asteroid belt, we don't actually have to do that! We normally imagine the asteroid belt to be a very thick belt full of space rock ready to hit anything in their path, but the reality is that the asteroid belt has a lot of empty space in it. Also the asteroids are not evenly spread out through the belt, they're often clustered together. So avoiding great clumps of them is even easier.

To get another idea of this, if you assume all the asteroids are on a flat surface, then a surface the size of the USA would have only 2000 asteroids. As of 2018, we would have about 10 times more towns and cities in the USA than asteroids.

While no Earthly organisms (human or otherwise, except perhaps unintentional bacteria) have been sent to Jupiter, several unmanned robotic missions have been used to study this giant planet (links on each page have more details). These first began in the 1970s with the Pioneer 10 and 11 and Voyager 1 and 2 flyby missions. Later orbital missions (e.g., Galileo and Juno) were added to study Jupiter in greater detail and for longer times. During these missions, the probes took visible-light pictures, made measurements of the magnetic field, charged particles (e.g., protons and ions), solid particles (mostly dust but also micrometeoroids), wavelengths and polarization of light to learn about the atmosphere composition and temperature, and cosmic rays and particles.

In addition, the craft would need to protect the passenger from extreme cold, heat, and radiation. The radiation is an especially challenging problem for Jupiter missions because the strong magnetic field leads to extremely intense radiation belts. These are like the Van Allen belts of Earth, but far more dangerous for humans. Another problem is muscle loss that occurs when astronauts no longer have to work against gravity to move. In less than 2 weeks, astronauts can lose 20% of their muscle mass, which makes completing their tasks difficult. It could potentially be even more dangerous for a trip to Jupiter where gravity is greater than on Earth.

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Astronauts on the ISS spend 2.5 hours exercising on specialized equipment every day just to combat this. All of that added "stuff" would greatly increase the mass which would need to be pushed away from Earth and complicate the design of the craft, both making the missions far more costly than unmanned missions. And despite all of these added complexities, there is still plenty to be learned without the costs and risks. In sum, manned missions to Jupiter just aren't worth the money, time, or effort right now.

As to going under/over the Asteroid Belt, doing so is (probably) technologically possible, but also highly impractical and unnecessary. For a long read and a bit of math (high-school level), the FAA has a good document. The relevant point for here is that changing a craft's orbit takes A LOT of energy. Since the Asteroid Belt is so tall, going around it would require a huge amount of fuel, a complicated trajectory utilizing gravity assists from other planets/moons, or both. But, as stated at the beginning, doing so is not needed to get to the other size. Contrary to Hollywood depictions, the Belt's asteroids are not packed close to each other such that constant adjustments are required to pass through. The asteroids are mostly small (< a few feet across) are spread out so thinly that (on average) each is surrounded by billions of miles of empty space.

Anyone curious about what would happen if a human did try to land on Jupiter, read here.

Nobody has ever been to Jupiter. We have sent several robot probes to Jupiter, but no human has ever been there.

There is no reason why you couldn't go underneath the asteroid belt (or above it), but there is also no reason to. The asteroids in the belt are so far apart that flying through it poses no meaningful danger to a spacecraft. If the asteroids were close together enough that you would have to weave around them, then we would not be able to see Jupiter from Earth.