Yes, Rio's reputation will improve following the Olympic Games because the actual games should not have any major security breaches or threats. Things such as athletes' clothes being stolen happens at every Olympic Games. This should not diminished the reputation of the city or country. People are making a bigger deal out of it because it is Rio.
Rio's reputation will continue to suffer following the Olympics. Although the city hoped for a boost, a series of mishaps, including the Zika virus and the stealing of an Australian team's laptops, have conspired to give the city a black eye. This will be difficult to recover from since this is a high profile platform.
Rio's reputation is likely to be worse following the Olympic Games. It is likely that crime and disease, especially the Zika virus, will be present during the Games. Unfortunately, any crime or spread of disease will likely be broadcast by the media and blown out of proportion. Rio's reputation will suffer following the Games.
Some teams are refusing to stay there, then we have things like thefts and robberies of tourists to consider. Rio has a reputation based on decades of police failing to control crime, or even pretending they want to. The pomp and circumstance will bring some good thoughts about Rio, but once it's over the city's reputation won't improve unless they make real changes.