How do I get Fathers illnesses connected to Coast Guard service Lighthouse keeping, worked under radar, asbestos on ship, 58/62 Cuba crisis?

Contact your local Veterans Affairs counselor to find out if they can help to qualify him for VA benefits. If you don't know who to call, contact your local VFW - Veterans of Foreign Wars to see if they have the name and number of the Counselor for the are where your father lives. You will need his discharge papers ( DD-214 or DD-two-fourteen for USNavy members). They should help you understand if he qualifies for medical care and if he might be deemed "disabled" due to exposure to chemicals or other service related conditions.
This takes time and some persistence but keep at it until you here from someone.
Best of luck.

I'd start with the internet and search for similar problems/issues with asbestos first. Then I'd contact a lawyer when you have something to show them. A lawyer could probably do the work of discovery, but you'll save yourself some money doing the leg work first. At least that's how I'd start.