The easiest option is to book a private taxi or transfer. You can find private transfers online for around US$ 100, or a private taxi will be maybe US$ 180. It's a 2.5 hour trip one-way and there are tolls, and obviously remember that a driver who takes you there has to then return, this is why it costs that much.
There is a bus from that area to Santo Domingo, but it doesn't depart from any hotels nor from the airport. So you would need to a) get a taxi to the bus terminal. b) Get the bus to Santo Domingo and c) get another taxi from the bus terminal in Santo Domingo to the ferry terminal. The bus itself costs RD$ 400 per person (about US$ 7) so if you don't have too much luggage and don't mind an adventure this is a cheap option. More information on the bus itself here:
https://www.expresobavaro.com/ . The bus is air conditioned and comfortable.
Another option to consider is hiring a car and driving yourself. It's a highway all the way from Punta Cana to Santo Domingo. The traffic does get very heavy as you get into Santo Domingo but you will be leaving the highway before the fun really starts, so provided you're experienced drivers you should be fine. All the main car hire firms operate in Punta Cana.
Final tip, Punta Cana is the name of the airport and Bavaro is the next town up, but very often these are confused especially online, so if you're searching for taxis or transfers or car hire, search using both names.
