Hi,
I usually take the ship sponsored tours. I pay a little more but have the comfort and backing of the cruise line and the tour desk. I also have the knowledge that they have generally done some type of research for me. All I have to do is read their brochure, make my selection, and pay them the money. They make all the arrangements for me and tell me where to be. If the ship is late, if it misses a port, if the weather is bad, etc. they take care of everything. In the event I am dissatisfied with a tour, (happened three times, one with NCL) out of about thirty or so, they have always refunded my money on-board the ship. I didn't have to deal with a poor tour operator directly. With most cruise lines, the tours can be booked on-line before you depart so you can enjoy your cruise with no-worries!
In many ports, I do my own thing. In Cozumel, a cab to Chankanaab is as good as any tour, in many ways, better! Sometimes, you can just do things on your own.
There are, however, many really good, independent tour operators out there that will offer you a lower price on the same tour you can purchase on-board. You may even have a better time! They also are aware of cruise ship schedules. Your purchases are somewhat protected with a credit card. Just to be safe, I would make sure the tour delivers me back to the ship at least 1 hour prior to the time I am supposed to be back on board, but that's just me. This is a good option for people who like to do a bit of research to save a few bucks. There may be tours available that the ship doesn't offer. You can also find these people lining the docks at the port holding signs selling these excursions. Do you feel like an adventure?