Just wanted to add some updated information since this is one of the first sites that comes up when I Google Monkfish. I was able to have quite a bit of luck not only catching this fish for my Grand Company Provisioning missions but getting quite a bit of exp in the process. I went from 27-32 in just a few hours. I was really just trying to pick up some HQ Monkfish for Grand Company Provisioning missions and was pleasantly surprised at the great exp.
I fished at Crescent Cove, using Shrimp Cage Feeders. I was using the best HQ gear for my level, and upgraded with gear from my armory chest as I leveled. Contrary to the information I found I actually caught a lot more Monkfish during daylight hours, I'd guess about 75% of the Monkfish were caught during the day.
None of these fish were able to be mooched, and mooching was not required to catch Monkfish.
Out of 233 catches over the course of three game days and nights, I was able to catch the following;
LVL 20 NQ Razor Clam – 43 (18.46%)
LVL 20 HQ Razor Clam – 9 (3.86%)
LVL 26 NQ Monkfish – 49 (21.03%)
LVL 26 HQ Monkfish – 13 (5.58%)
LVL 22 NQ Blowfish – 33 (14.16%)
LVL 22 HQ Blowfish – 13 (5.58%)
LVL 24 NQ Saber Sardine – 26 (11.16%)
LVL 24 HQ Saber Sardine – 10 (4.29%)
LVL 03 NQ Lominsan Anchovy – 32 (13.73%)
LVL 03 HQ Lominsan Anchovy – 5 (2.15%)
Considering that the bait only cost about 1500 gil I'm pretty happy with these results. Monkfish are selling for 1500/3500 on my server right now and Saber Sardines are selling for 1000/2500. I made enough gill to fully outfit my fisher, with the added bonus of having the gear available for my other DoL classes now too!
On a side note I went to grinding for exp after this, and was able to go from 32 to 36 very quickly fishing in Ceruleum Field using Syrphid Baskets. Two stacks of baskets (less than 700 gil) along with Grand Company manuals was all it took. Grip Killfish are 100% catch there, and bite within 2-5 seconds of casting. The NQ ones give 3200 exp and the HQ ones give 9500 exp. I caught a total of 165 (83.76%) NQ and 32 (16.24%) HQ. Unfortunately those are worthless on the Market Boards