Reggae Beach Bar on Cockleshell has free WiFi. Can't say for sure if other places have WiFi but you'll probably find at least one or two places on each of South Friars and Frigate Bay that have it.
As for the water taxi quotes you got I can't give you specifics as I've never had a need to take a water taxi from the marina by the cruise ship dock to those beaches, however, those prices seem about right. Water Taxis use a lot of gas and gas here is expensive. Those beaches are further away from the marina than you think. Cockleshell is the furthest (probably about 20 - 25 minutes by fast boat).
If the cost concerns you take a land taxi - the one way fares for up to 4 people are as follows;
Frigate Bay - $12US
South Friars - $16US
Cockleshell - $28US