In early June 2006, we experienced Lows in the upper 70s to 80 and Highs in the mid 80s every day for the week we were there.  The forecast almost every day was for partly sunny to partly cloudy skies with a 30-60% chance of scattered or isolated thunder showers.  There were generally light to moderate sea breezes every day. Humidity ranged from moderate to high. There was a tropical storm, Alberto, in the Caribbean and Gulf of Mexico, but Grand Bahama only saw some showers for a day.  The impression I got is that this is typical weather for the latter half of May and the first part of June.  Unless a tropical storm is bearing down on the Bahamas - not a huge chance for any given week in June - it is a good time to visit, with comfortable temperatures, quite a bit of sunny weather and water temperatures in the upper 70s to low 80s.