You can make a device to save the star. All you need to do is generate research points at that colony. At a couple of points you will get another news report saying how many more points you need.
The amount of research needed depends on the size and development of the colony and only research from that colony will work. Usually setting research to max will give you enough points with some time to spare. If things get close you can always transfers some money to the colony to double its production for a turn or two.
I can’t remember exactly but I think if the star blows up your planet becomes radiated and loses a good chunk of its population.
