Before we think of ways to solve this problem, we must acknowledge two things. First, we must realize that it is probably not possible to eradicate all structural unemployment. So far as we know, this has never been accomplished in any modern society. Second, we must also note that economists say that it is important to have some structural unemployment. If we have no structural unemployment, it means that our economy is not progressing. We are not discarding any old and obsolete jobs for newer ones. Just as an example, if we never had any structural unemployment, we would not have gone away from using horses to using automobiles (because all the horse-related workers became structurally unemployed).
All of that said, there are things that could be done to reduce the amount of structural unemployment and to try to ensure that people would not remain structurally unemployed for a long period of time. There are two things that would be the most likely to help. First, we would want to ensure job creation in our economy. This would include doing things like subsidizing research and development so that new industries could arise. It would also include making it easier (less red tape imposed by the government, for example) for businesses to hire new workers. Second, we could try to make sure that American workers had a variety of marketable skills. When people become structurally unemployed, we could provide them with retraining opportunities so they could get new jobs.
Of course, these things are more easily said than done, which is one reason why it is hard to eradicate structural unemployment.