It's generally safe to leave the resort in Punta Cana; it's recommended that you don't do it alone and instead book a safe shuttle to Punta Cana. It's safe to leave the resorts of Punta Cana, as long as you go with a guided tour. Not only will experienced tour guides take you to the best destinations, but they will also help you navigate the country safely. For greater security, book tours with the resort's travel agents.
They only use reputable companies, so you don't have to worry about the credentials of a tour guide. Off-site tours and excursions are easy to plan when using a Punta Cana resort as a base. One way to organize a client's schedule is to alternate days at the resort with days full of experiences outside the resort. While Punta Cana is prone to some of these events, they are infrequent and will often be notified in advance.
However, sometimes tourists like to take day trips from Punta Cana to Santo Domingo to see the capital of the Dominican Republic. As an Airbnb host, I schedule a lot of taxi trips in Punta Cana and the amount you pay for a taxi varies a lot depending on the distance of your trip. The threat of a storm shouldn't stop you from enjoying your trip to Punta Cana; just remember to use common sense in the face of impending storms and listen to earthquake warnings. Living Punta Cana Tours (LPC Tours) adds a boat ride with lunch on the Chavón River as part of its excursion.
Take to the skies and fly over Punta Cana on a hot air balloon ride with Dominican Balloons, the only balloon travel operator in the entire Caribbean. The ocean that surrounds Punta Cana has the possibility of harboring organisms such as sea urchins, the Portuguese Man of War, Cassiopea, stingrays and lionfish. In fact, Punta Cana is significantly safer than other cities in the Caribbean, so most of the concern is unjustified.