When we went in April this year, we drove ourselves and asked around at the car park to find two men with a boat who took us over the lake to the island. However, according to a poster a couple of months ago, that boat wasn't working when they visited, so I don't know the situation now.
Olivier Messmer's company is called Ecotour Barahona, and the website is
Ecotour Barahona, the ecologic tourism in Dominican Republic. They do a one-day tour to Lago Enriquillo out of Barahona. I haven't used this company personally, but other posters have, and a search of the archives will give you more details.
It would be just about possible to drive yourself there and back from Santo Domingo in a day, but it would be an incredibly tiring day with a very early start.
I second Dolores' recommendation of a hat, water and suntan lotion. I think it must have been the hottest place I have ever been to.
