There are a few factors that will tell you a perennial would benefit from being divided:
It has outgrown the space. If the plant is crowding its neighbors or spilling into a pathway, you can divide it to solve the problem. Replant one of the smaller divisions in the same spot. Discard the rest or plant them elsewhere in the garden, where they'll have space to grow.
It has taken on a "doughnut" appearance. If there's a dead spot in the middle of the plant, with fresh growth circling it, you need to dig it up, discard the "doughnut hole" and replant the live parts.
It hasn't been blooming as well as in past years. Diminished flowering can be a sign of a plant in decline. Splitting it will give the smaller portions fresh life. Fewer blooms can also result from lower light levels, however; so before you divide, take a look around to see if the plant is receiving more shade than in past years due to neighboring plants or trees growing larger and shading it.
To review what time of year to divide perennials, see "The Best Season to Divide Perennial Plants."