There are several things that could happen when sailing during hurricane season:
If a storm is predicted to affect routes or ports on a cruise, the captain will try to reroute the ship to avoid the storm. This could mean missing a port or arriving late/leaving early. This is the most common effect of storms. Because the changes are due to weather beyond
DCL's control, you may not be offered any compensation (beyond refunds for any ship-booked excursions in cancelled ports), but addition onboard activities will be scheduled if you miss a port.
If a storm changes course unexpectedly or cannot be avoided, the ship may experience turbulence. This is rare, but has happened.
If a storm will be hitting your home area or your arrival/departure city (or areas in between), your flight may be delayed or cancelled or you may run into freeway delays. This is why it's recommended to plan to arrive at least a day before your cruise, and schedule return flights no earlier than afternoon.
If a strong storm is expected to hit the port on your arrival or departure day, the ship may divert to a different port. In those rare cases,
Disney Cruise typically provides transportation to/from the original port, but that is not guaranteed.
If a strong storm is expected to hit the port on your departure day and/or to stay in the area where the ship is planned to travel, the cruise may be cancelled altogether. In that case, you will be given a full refund and usually a discount on a future cruise.
Most cruises during hurricane season run as expected. Some will have adjustments to avoid a storm. It is unlikely that the arrival/departure ports are affected enough to cause cancelations or diversions, but that is a possibility. Travel insurance may compensate you for needing to change your flight, staying at a hotel because you can't get home, or missing the cruise because you can't get to the port.