I think both airport buses and taxis are safe and OK.
Buses from GIG (they stop at multiple places):
Line 2018 to Barra da Tijuca (Alvorada) via zona sul 15R$
Line 2918 to Barra da Tijuca (Alvorada) via Linha Amarela 15R$
Line 2145 to airport Santos Dumont via centro 13R$
Line 2101 to airport express to SDU 13R$ (not sure if this exists, I have taken line 2145)
Taxis are allowed to use the higher rate "2" on Sundays and holidays.
Should be 60R$ or more to Copacabana/Ipanema