Carrie,
I hope I'm not too late to help you out with this.
An electric fence is the way to go. You can run a simple single strand of electrified wire with a solar panel charger that will do plenty to keep the bears out. I work in the bush in Kodiak and we always surround our rafts, atv's, etc with electric fence. For some reason plastic and rubber are like candy to bears.
Once you put your fence up tie some surveyor's tape at different locations on the wire. This serves two purposes. For one, it allows you and anyone else to see where the fence is. Two, bears are very, very curious and they'll be drawn to the flagging. When they investigate the first thing they use is their nose. All it takes is one zap on the nose and they will not come near that coop again!
I wouldn't bother with barbed wire. For one, it will snag everyone and everything that comes by and if it's electrified as well an animal can get caught in it and shocked to death. Smooth wire allows them to feel the electricity but still get away. Also, Alaska's bears have extremely thick layers of fur and the barbed wire would not get through to the skin to cause any damage - only your fence would end up damaged.
A lot of people seem to be raising chickens on Kodiak and the electric wire keeps out even the most determined garbage bears. They are really quite simple to use and can be obtained at almost any farm/feed store.
Best of luck!