There is a well defined, hard-surfaced and well used running path pretty much along the entire beachfront area. Very conducive to running and you will be in good company. Gets pretty crowded on the weekends and you may have to duck between the folks who stop in the middle to take pictures, etc ...

Safety wise ... well, nowhere is perfectly safe. But in my experience, I have ran both in daylight and into the evenings when it is cooler. Gym shorts, t-shirt, some $$ tucked in my shoes. (No fancy ipod, etc). No issues so far.

One thing to note is traffic. Not sure how sensitive you are but the road right by Copacabana/Ipanema can be pretty crowded with cars/trucks/etc. I hate the exhaust. My best running experience was when I was staying further west outside of Rio city proper, in Barra. There is a nice LONG beach and running path right beside it. Not as much eye candy though ...