The only cure for this is, of course, age and experience. You have to be careful not to let them cling to you. Be their friend, and try to get their minds off their missed kin at home and onto learning English and American culture, which is why they are here. Serious problems are a fact of life everywhere. Some examples of serious problems can be medical problems, landlord problems, legal problems, immigration problems, or something as bad as suicidal tendencies.
People are people.
Being able to spot serious difficulties isn't hard. They will manifest themselves in the student's demeanor, or the student will just tell you. Often you will be the first person in this country to which the ESL student will bring a real problem. All serious problems ought to be referred to a specialist. In most cases, cultural adjustment is the cause for an ESL student's problems.