Gently wash the apricots and cut them in half. Discard all but 3 of the pits.
Wrap the 3 saved pits in cloth and smash them with a meat mallet or the bottom of a small sauté pan, just enough to break them open. Wrap the crushed pits in cheesecloth and tie it with butchers twine. (Use two layers of the cheesecloth, and no more than 3 pits.) Adding this bundle will add a lovely and subtle almond flavor to the finished preserves. Set aside.
Add the water, sugar, and basil leaves to a medium-sized, heavy-bottomed pot (do not use an aluminum or un-coated cast iron pot).
Heat just until the sugar has dissolved and then add half of cut apricots and the cheesecloth bundle of crushed pits. Simmer for 30 minutes, stirring every 10 minutes or so. (I don't like to mash the fruit, because I like to end up with big chunks in the preserves.) Add the remaining half of the cut apricots and simmer for another 30 minutes, again, stirring from time to time.
Remove the cheesecloth with the crushed pits, and let the preserves cool completely before refrigerating.