A recent United States Supreme Court case, Norfolk & Western Railway Co. v. Ayers, held that the plaintiffs, who were already sick with asbestosis from asbestos exposure, could also receive damages for emotional distress based on the fact that roughly 10% of patients with asbestosis develop and die from mesothelioma. The Court excluded those who had been exposed to asbestos but were not yet sick, stating that to do otherwise would be to open the claims up to unlimited and unpredictable numbers. The Court also found that there needed to be both a serious and genuine fear of developing a more serious disease, and that this might preclude liability on this issue in the future.
Therefore, since you are not currently suffering from any asbestos related illness, your best plan right now is to document any asbestos exposure that you suspect you have had, where you were exposed, and for how long.