Just how crowded is the asteroid belt?

Let's do the math!

There are about 3500 - 4000 or so cataloged asteroids with diameters over 1 kilometer. The asteroid belt is a band between 2 - 3.3 Astronomical Units wide, give or take, but the asteroids are bunched up into families so are not uniformly spread out in this vast volume some 100 million miles wide and perhaps another 20 million miles thick.

The volume of this space is about 4 million trillion cubic miles. The average distance between the asteroids would be about 100,000 miles. But there are likely to be lots of smaller objects too. Both the Voyager 1 and 2 spacecraft passed through the asteroid belt. They were not hit by anything except interplanetary dust which is very common in the asteroid belt, and that is your biggest problem to worry about.