Crandon Marina mooring field is full with a waiting list that is years long. Dinner Key has a new mooring field at $300 a month with a launch service that runs about hourly or so, but not after 6PM. But it is totally exposed and has no real security during the day and none at night. The Coconut Grove Sailing Club would cost you more (if they would have space for your boat's beam), about $500 a month, but it has a 24 hour launch service which is very fast, is very well protected, has 24 hour security, and it is a very short walk to the heart of Coconut Grove. You don't have to be a member to moor there. It is were I have elected to keep my boat, and I had my choice of places.
As to Miami, it is very easy to get to from NY and there are a number of places you could sail to, even if you just had a long weekend. No other place on the east coast of Florida can say that. Here you could gunk hole like you do on the east end of the Long Island Sound. Of course, Bimini is only 45 miles away, but you'll need to wait for favorable winds, which sometimes don't appear when wanted.