Every night. Without getting out an astronomical program, I'd say the moon rose somewhere round midnight early in the voyage. By 15 April, it was rising just before dawn. The first nights would have been quite bright, with a good half moon.
Moonrise was always so late that it didn't interfere with celestial navigation.
A decent moon that was high in the sky before midnight would have saved the ship. I've never seen a berg, as they are rather scarce in my home waters, but I have seen a major reef from several miles off in bright moonlight.
The Titanic story is famous for the number of 'ifs'.