Are there pay phones or internet cafes on Molokai? What is the best way to
stay in touch with the mainland?

There several pay phones in or near Kaunakakai.
In front of the post office,
In front of the American Savings Bank,
In front of Misaki’s Grocery,
In front Hawaiian Telcom office,
At One Ali’i Park

Cell phone service is somewhat spotty on the island, with best signal strength available from Verizon or At&t. Pretty good signal available from one mile east of Kaunakakai and continuing past the airport to the Kauluakoi turnoff at mile 15 west. Some locations on the east end after mile 13, pickup signals from Maui.

The closest thing to an internet cafe on Molokai is the lanai at Coffees of Hawaii in Kaulapu’u. Bring your own laptop computer, they provide free, high-speed, wireless access. There is also WiFi available at Molokai Burgers and Molokai Pharmacy. In addition, the library in Kaunakakai has a few computers that are available for patrons.

Is high speed internet available to purchase if you rent a property on Molokai?
Do you know if it’s DSL or cable?

If my husband accepts the job I’ll be running an internet business and will need reliable, fast service. Any help you can give on this subject would be much appreciated!

DSL is available in and around Kaunakakai. High speed cable access is available most everywhere on the island. While the cable offers somewhat higher speeds, it is susceptible to downtime, especially during inclement weather. On the other hand, my DSL connection has never been down even when my land line voice connection failed during a huge rain storm.