The Atlantic hurricane season runs from 1 June to 30 November, and cruise ships keep sailing through it. The relevant point for a passenger is not that storms exist but that a ship is mobile in a way a hotel is not: it can leave, and the itinerary is the part that absorbs the change.
When a system develops, the practical outcome is usually a reordered or substituted itinerary rather than a cancelled cruise. A route planned for the eastern Caribbean may run west instead, calls may be swapped for others at a similar distance, and a departure may be brought forward or held back so the ship can clear the forecast track. The ports themselves may be unaffected; the change is about where the ship can safely be on a given day.
Two consequences follow. First, an itinerary booked inside the season should be read as a likely shape rather than a fixed list of islands, so the cruise is a poor way to guarantee one specific place. Second, the risk is uneven across the season and across the region, and official forecasting bodies rather than travel sites are the source to check for any particular date.