Variable. It depends somewhat on the fruit (how chewy it is) and how well your child chews. Many toddlers do quite well with chewy soft raisins at 15-18 months for instance and others who cannot chew well for another year.
2-3 years. Babies are too young for dried fruits (babies eat pureed or mashed up foods, after age 6 months). Dried fruits are tough to chew, so a toddler who has baby molars and is actually chewing foods might be able to handle dried fruits. A younger toddler may just hold the dried fruit in his mouth for the sweet taste, and either swallow it whole or spit it out later (may be a choking risk sometimes).
3 years. Wait at least 3, preferably more, years, as they can be a choking/aspiration/foreign body in nose hazard.