The best time to cross is in advance of a cold front where the wind shifts to the south or southwest. But better make damn sure you're in Bimini before the wind clocks to the North/Northeast, as the Gulfstream can turn hellacious in the actual front itself. In fact, your timing of this question is ideal...there's a front arriving tomorrow night. Just look at this forecast, remember that leaving from SE Florida, you're generally going to have to make southing to counter the effect of the 2-3 knot current pushing you north.
Before the front...when I would cross. Start as far south as possible but by all means BE OUT OF THE GULFSTREAM BEFORE THE FRONT ARRIVES. Motor sail or just motor!!!
SOUTH SOUTHWEST WINDS 5 TO 10 KNOTS BECOMING WEST
SOUTHWEST IN THE AFTERNOON. SEAS LESS THAN 2 FEET. DOMINANT PERIOD
8 SECONDS. INTRACOASTAL WATERS A LIGHT CHOP.
Now the front arrives...notice how things are starting to get dangerous. West wind is good *nearshore* as it lays the seas down...however offshore, westerlies tend to be strong and GUSTY, causing steeper seas.
WEST WINDS 10 TO 15 KNOTS BECOMING NORTH
NORTHWEST 15 TO 20 KNOTS AFTER MIDNIGHT. SEAS AROUND 2 FEET
BUILDING TO 2 TO 4 FEET WITH OCCASIONAL SEAS TO 5 FEET AFTER
MIDNIGHT. DOMINANT PERIOD 5 SECONDS. INTRACOASTAL WATERS A MODERATE CHOP.
ISOLATED SHOWERS AFTER MIDNIGHT.
All hell breaking loose as the front arrives...better pray you're in safe harbor (not even anchoring...a marina) when this slop arrives.
NORTH WINDS 20 TO 25 KNOTS. SEAS 9 TO 11 FEET WITH
OCCASIONAL SEAS TO 14 FEET. DOMINANT PERIOD 7 SECONDS. NORTH NORTHEAST
SWELL 3 FEET IN THE AFTERNOON. INTRACOASTAL WATERS A MODERATE
CHOP. A SLIGHT CHANCE OF SHOWERS.
NORTHEAST WINDS 15 TO 20 KNOTS. SEAS 9 TO
11 FEET WITH OCCASIONAL SEAS TO 14 FEET SUBSIDING TO 7 TO 9 FEET
WITH OCCASIONAL SEAS TO 11 FEET AFTER MIDNIGHT. NORTH SWELL
3 FEET IN THE EVENING. INTRACOASTAL WATERS CHOPPY IN EXPOSED
My usual path was to leave port everglades and motor as far south...usually to about Baker's Haulover Inlet. To avoid the stream, we would hug the shore in about 15-20 feet of water trying to stay inside the dive buoys lining the coast. Go as close to shore as you're willing to. THen make a sharp left and sail *HARD* east...motoring as fast as you can. This would generally cause us to land about 4-5 miles north of Bimini due to current. We'd leave at night and hope to arrive 10 hours later at the banks. Then we'd sail or motor back south, casting lines to try and catch a mahi. Its an "s" shaped route. When we arrive, into the marina, clear customs....then pass out from lack of sleep as the front arrives and wind howls. Wake up, clean the fish we caught earlier, then drinks at the bar as the front moves off and we plan our next passage further east. Not much to do or see in Bimini beyond a long weekend. Waters are too shark infested to dive nowadays.