You can save some money when you decide to take an independent tour from the cruise line. The plus is that the tour isn't as large and just for your party or a tour that is not offered through the cruise line. There is a level of service that comes with this compared to a large group of people being shuffled through.
There are ways to mitigate issues that do arise. Here are some tips:
Find a tour company that will cater to the cruise line- pickup and drop off at the cruise terminal.
Know and set your watch an alarm time when you need to get back to the ship- local time vs ship time.
Research your tour and allow time to arrive early for your meeting point, traffic delays, and bring money in USD or foreign currency for tipping.
Bring a copy of your passport & don't forget your SeaPass card.
Insurance for any accidents that do happen (fall, dental and medical problems, etc. )
Check for reviews of the tour company- Reputation, Reviews on Google or TripAdvisor?
Make sure you have the day of tour contact name or phone number.
Keep this in mind.
As you are responsible to get to the next port of call if you do miss the ship. The insurance may or may not cover you.
In recent news, a guest passed away after taking an independent snorkeling excursion in the Bahamas. The cruise lines are aware of these accidents do happen on tours and they do try to support and assist in any way they can.