The Netherlands Antilles, with cash flowing steadily from banking centers in Amsterdam and Rotterdam, is a favorite financial center for investors seeking a low profile.

In Willemstad, the sunny Caribbean capital of the Netherlands Antilles, a banker ushers an American visitor through a hotel casino and into a dining room overlooking the harbor. During refreshments, the prospective customer says he expects a six-figure cash windfall soon and would like to bring the money “quietly” into…