Deep Sea Fishing is not particularly dangerous at all, particularly as compared to other action and watersports. In the many years around the docks I have not see any major injuries. The boats and captains are forced to comply with strict United States Coast Guard regulations, that dictate required safety equipment, for boats and training programs, physical testing, and drug testing for captains -and they do keep an eye on use -we also keep an eye on each other.
The most common risks of deep sea fishing are actually slipping on fish, hook in the hand, sunburn and dehydration, The last two are easily preventable with a bottle of sunscreen and a bottle of water. Less common risks include, scrapes, scratches, and bruises, allergic reactions, and unlikely but possible: falling overboard.
Children are required to wear life jackets and it is recommended that adults who are not strong swimmers wear life jackets as well. If you are bringing children aboard, contact the captain ahead of time to make sure he has appropriate life jackets for the size of your child.