|Credit: Memory Alpha.|
In their only appearance, a group of them (Bynars always come in pairs) attempted to hijack the Enterprise-D while they were upgrading the ship's computers and the crew had evacuated. After being twarted by Picard and Riker, the Bynars revealed that the goal was to use the Enterprise to save their species - A nearby star went nova and the electromagnetic pulse would have fried their supercomputer, which presumably would have been fatal to the Bynars. Their plan was to use the ship's computer as a backup until the danger passed, then they would restore their own. Unfortunately, the Enterprise was late in at arriving for its refit and so the Bynars had to steal the ship. They were also working under the assumption that their plea for help would be turned down.
The Bynars are one of those one-off races in Star Trek that should have been used more. It would have been interesting to see one or two of them as crew members on the Enterprise.
Outside of Trek, a Bynar-like race would be cool in both prose and RPGs.