In case you bring your laptop on the cruise, and you want to use it. There is wireless internet at the food area of La Gran Plaza mall, also at McDonalds in the golden zone. Some of the hotels have it also, some is free off the air and some might require a password.
Although you probably will not go there, but they have it also at the Mazatlan airport.
If you did not bring your laptop there are plenty of cyber cafes around the city. In the terminal building at the dock, there is one, but <I think it is about 5 us an hour. Outside across the street inside a supermarket, they have computers and it is about 1 or 1.50us and hour. In the golden zone it is probably about 4 to 5us an hour. >But still it is less then on the ships....
Hope this helps.