The various moons are on orbits that are mostly a lot more than 1km apart from other moons. The orbits are stable, so the moons just keep going round Saturn on the same orbit. The moons are far enough apart that they have negligible gravitational influence on other moons. So the moons never get close to each other and don't collide. There are moons, like Daphnis and Atlas that can come with 1000km of each other, but they are very small. No moons come within 1km of each other.
There are some exceptions to this: Janus and Epimetheus orbit at nearly the same distance from Saturn. If they stayed like this they would collide within each other within about 4 years. However as one moon starts to catch up with the other, it is pulled forwards, causing it to move out, and slow down. Letting the other moon move inwards and faster. The two moons essentially swap orbits. Although there is only about 50km between the orbits, the two moons never approach closer than 10000km, so are not at any risk of collision.
Moon collisions can take place, one theory on the origin of the rings is they they were formed from a moon that broke up, either from a collision or by Saturn's gravity (It's not the most popular or recent theory). Collisions have never been observed, nor are any forecast to occur for as long as we can predict.
Of the 62 moons, remember that some of them are extremely small: a few tens of metres across, and there is no clear division between "small moon" and "large clump of ring material"