Premium Auto Ônibus operates safe air-con buses from the international airport along several itineraries. For the Zona Sul, take bus 2018, which heads south through the bairros (districts) of Glória, Flamengo and Botafogo, and along the beaches of Copacabana, Ipanema and Leblon to Barra da Tijuca (and vice versa) every 30 minutes from 5:30am to 11pm. It'll stop wherever you ask.
The trip takes one to two hours depending on traffic. Heading to the airports, you can catch this bus from in front of the major hotels along the main beaches, but you have to look alive and flag it down.
If you're going straight to Barra, it's faster to take bus 2918 instead. There's also a bus that links to Santos Dumont Airport (bus 2101) and a line to the bus station (2145).