Here comes a small list of Caribbean Banks that I have been looking at in 2016, it is not a complete list but contains only banks that I have found to be interesting for my business.
You may use them at your own risk, try contact them directly by phone or e-mail and see how they can work with your business and if they accept remote account opening.
First Caribbean Bank
Internet Banking : LogOn - CIBC FirstCaribbean International Bank
They have a branch in the BIV and St. Maarten
Phone:
1-800-744-3242
Royal Bank Bahamas
Online Banking
Branches in the Bahamas and Turks & Caicos Islands
Seems to be a small bank, chances are they will be willing to open an account for you remote if you speak to them.
If you stumble upon a bank in the Caribbean or rather a website that claims to be a bank but you have a feeling it is not the case then you want to visit the Eastern Caribbean Central Bank's website
Eastern Caribbean Central Bank Financial Institutions to check if what you have found is a legit bank operating in the Cribbean or if they are some shady operation.