At the nursery we divide clumps every two or three years to increase our stock. We perform this function in the spring shortly after the "bullet" stage...when the divisions have just begun emerging from the ground.
Try to salvage as much of the root system as practical. After the clump is dug, we wash the roots so that we can see where to divide....otherwise you may cut too many roots.
Most hostas have a basal plate of hard tissue between the foliage and the roots. We use sharp kitchen knives to perform the actual division...using a size that fits the size of the hosta. Try to cut through only this basal plate area...and avoid cutting off roots.
For hostas without this hard tissue basal plate, we have found it best to NOT use a knife to divide. Instead of cutting, use a gentle twisting movement to separate the clump. For hostas without a solid basal plate, this method insures that fewer roots are disturbed.
We generally transplant immediately after division, although sometimes we'll allow the plant to "harden off" for a day or so. The hardening off process may reduce chances of rot....but we rarely have this problem even when transplanting immediately.