If you live in Luxembourg you usually do not need a Caribbean visa.
Luxembourg is part of the Schengen area. Do you have the nationality of Luxembourg? Or do you have a valid Schengen visa or a residence permit for Luxembourg? Then you do not need to apply for a short-stay Caribbean visa to travel to the Caribbean part of the Netherlands (Aruba, Bonaire, Curaçao, Saba, St Eustatius or St Maarten).
In case you do need a short-stay Caribbean visa, please make an appointment at the Netherlands embassy in Luxembourg.