We split our hostas throughout the year and quite frankly they are not at all fussy about it! However, the most practical time to split them is in the early spring when the shoots are just appearing like small pencils through the ground. The main reason for this is that it is far easier to see where the divisions are and of course you do not risk damaging the leaves.
Occasionally if you allow a hosta to get 'too big', then it can suffer from crown rot, when the middle starts rot, whilst the outside remains healthy. So, if you are unsure, then split away. However, bear in mind that Ann & Roger have a specimen of Snowden which has remained undisturbed for about 30 years and looks extremely content! So if your hosta appears to be healthy, then don't feel that you have to split it.
Work out where the split needs to be and then take a knife, a hacksaw or a spade (depending upon the size of the plant) and cut through right to the bottom of the roots. The hosta will not suffer, indeed this often makes a plant throw up new shoots and you have an extra hosta to plant elsewhere or to give away. In America hostas are known as the friendship plant because of the ability to share in this way.
Hostas in pots can, of course, become pot bound and require either splitting or moving to a larger pot. If your hosta is in a container with a fat middle but a narrower neck, you may need plenty of patience and ingenuity to extract the hosta from the pot.