Take a look at St James Club, Antigua. Family friendly, especially if you go for one of their villas as you get a separate bedroom and living area with a small kitchenette.
It has kids clubs and entertainment, which are very good when they run - but do operate on Caribbean time. (I.e. not necessarily at the time advertised, just whenever they get round to it.) All part of the more relaxed atmosphere! However it's not got much nightlife, which might be either a pro or a con.
There is a seasonal (March - October, IIRC) seaweed issue across Atlantic facing resorts in the Caribbean. Last time I checked, the family resorts were on that side of Antigua. However, St James' has a second beach onto a bay, which is sheltered and is good for small children because it's shallow and the water is calm.
Antigua is lovely. I've been there a few times and always felt safe. The people are very friendly, and there are a variety of excursions you can do round the island, from ziplining, to the historic harbour, to luxury yachts, to museums, to shopping, to parties, to cooking, to nature reserves, to conservation efforts, etc.