I would also cut it down. You figure while the fronds were there they were helping the plant make food, which it stored in its roots. If the stem is already brown it's not photosynthesizing any more, so may as well cut its losses and let the plant get ready to put up sprouts next time around.
As for water- not sure where you are, but I would
water minimally. For me that means almost never, since I get rain every so often, YMMV.