There are more than 100,000 asteroids larger than 1 kilometer in diameter, but these objects are distributed within the huge volume of the asteroid belt. Their average spacing is several million kilometers. Collisions are thus extremely rare; an average 1-kilometer asteroid suffers one collision every few billion years, or maybe one or two collisions over the lifetime of the solar system.
Spacecraft pass through the asteroid belt with virtually no chance of a collision, and in fact considerable effort is required for a close flyby of even one asteroid, such as the Galileo spacecraft flyby of Ida in 1993. The spacing is also so large that seen from one asteroid, even the nearest 1-kilometer asteroid would likely be too faint to be visible without a telescope.
An illustration of how to calculate spacing and collision probabilities in the asteroid belt can be found in The Planetary Systyem by D. Morrison and T. Owen, on page 129 of the second edition.
Followup posted September 2, 1997
Even if there were 100,000 sizable asteroids (more than a few kilometers in size) in the asteroid belt--and the real number is quite likely about ten times less--the average separation between them would be about five million kilometers. That is more than ten times the distance between the earth and the moon. If you were standing on one of those asteroids and looked up, you would not see a sky full of asteroids; your neighbors would appear so small and dim that you would be quite lucky to even see one, let alone hundreds. I would not even call crossing through such emptiness terribly "interesting," which is fortunate for the space probes like Galileo and the Voyagers that have had to do just that.
Tom Gehrels of the University of Arizona is a veteran asteroid hunter; using the Spacewatch Telescope on Kitt Peak, he and his colleagues find roughly 20,000 objects a year, many of them uncatalogued asteroids. He adds some additional information:
Some scientists were seriously concerned about the possible high density of objects in the asteroid belt, which lies between the orbits of Mars and Jupiter, when the first robotic spacecraft were scheduled to be sent through it. The first crossing of the asteroid belt took place in the early 1970s, when the Pioneer 10 and Pioneer 11 spacecraft journeyed to Jupiter and beyond. The danger lies not in the risk of hitting a large object. In fact, such a risk is minuscule because there is a tremendous amount of space between Mars and Jupiter and because the objects there are very small in relation. Even though there are perhaps a million asteroids larger than one kilometer in diameter, the chance of a spacecraft not getting through the asteroid belt is nearly negligible.
In some ways, the asteroid belt is actually emptier than we might like. In the early 1990s, the National Aeronautics and Space Administration wanted the Galileo spacecraft to encounter an asteroid while it was passing through the asteroid belt on its way to Jupiter. But it took some effort to find an object that was located even roughly along Galileo's path. Special targeting was required to reach this object, but the result was the first close-up view of an asteroid, the one called Gaspra.
The number of objects in the asteroid belt increases steeply with decreasing size, but even at micrometer sizes the Pioneer spacecraft were hit only a few times during their passage. That is not to say that asteroids cannot pose any danger, however. It is worth noting that for a large planet like the earth, over a long period of time, there is an appreciable chance of being hit. This hazard comes from the fragments of mutual collisions in the asteroid belt; after their break-up, some of these fragments move towards the earth under the gravitational action of Jupiter. The largest of these pieces of debris is about 12 kilometers in diameter.