Vacation in January

Hello! We are thinking of taking a family vacation next December/January in the Caribbean. There will be probably three families with kids ranging in age from 4 to 16. Would a resort be better? Airbnb? We would love to get out and explore the local community so which island might be the safest? Also looking for somewhere that has beautiful waters/beaches and fun activities but also budget friendly. My family in particular would be 2 adults and 3 kids 8 and under and would like to keep the price around $5,000-6,000. Any and all recommendations welcome!