If you stay on the hook, don't go to restaurants or bars much, and don't have any significant boat repairs to pay for, $1,200 per month might do it. Although, baby food and disposable diapers aren't cheap. On the hook, you'll need a good size house battery bank, and all the gear to keep it charged.
$1,200/mo is only $10 per day, and food is significantly more expensive in the eastern Caribbean, unless you can make fish you catch and local vegetables and fruit a major part of your diet. And even the latter are somewhat expensive unless you buy them where they're grown in quantity, like Dominica or St. Vincent. For example, the Dominicans come to Nevis once a week in a small ship, and sell their fruit and vegetables for half of what the local vendors charge, and probably twice what they'd charge in Dominica.