2016 saw some Zika infections reported in the Bahamas, but generally there are no exceptional health concerns for visitors.
The tap water is fine to drink, but can be unpleasant in places (where the locals often resort to wasteful bottled water).
Nassau and Grand Bahama have modern hospitals with emergency rooms open 24/7, but free care is only provided to legal residents. The Out Islands are serviced by small government clinics, usually found off the Queen’s Hwy in the major settlements and open 9am to 5pm Monday to Friday. All will have 24-hour emergency numbers posted outside, although serious conditions will have to be treated in one of the major centers.
While accidents and illness are always possible, there's no need to take out anything beyond the standard health and travel insurance when visiting. Consult your insurer for further details.
Travelers staying over one year and arriving in the Bahamas from Angola, Benin, Bolivia, Brazil, Burkina Faso, Colombia, Cameroon, Democratic Republic of Congo, Ecuador, French Guiana, Gabon, Ghana, Guinea, Liberia, Nigeria, Sierra Leone, Peru and Sudan must provide a certificate of yellow-fever vaccination.