I'm pretty sure leatherbacks are protected, although maybe this was lifted, I haven't read or shown interest in these things for quite some time.
It depends on what kind of turtle you are talking about, some are endangered, some allowed to be captured through certain parts of the year. Sea turtles status is all over the place, as soon as the specific breed is off the endangered list, then they are open to capture again, and then back on the endangered list within the year, it's pathetic.
In my opinion, they should simply have an outright ban on the capture and/or slaughter of these animals. There is a reason this country has no wildlife of any sizable animals, and that's because Dominicans tend to be extremely ignorant when it comes to any kind of conservation, either of their land, what lives on it or even in the ocean around it. As soon as you tell a Dominican he can kill something for profit then he will simply kill, kill, kill until he can't anymore. A bit like the fox in the chicken run scenario.
Simply on moral standards these animals should be left to do their thing, I don't think there is another animal less defenseless that I can think if. They deserve a little respect, they were here long before us.
But in order to answer your question specifically, you'd need to keep up to date with what is at present endangered and what months you can capture them, if any at all. Like I say, I've not shown any interest in any of this for quite some time, hopefully laws have been stiffened, I'd love to be utterly wrong on this one.
really don't watch this unless you are heartless, it will ruin your day!
http://m.youtube.com/watch?v=VUC5vTQp-nM