It is my understanding that it is difficult to fish on a sailing ship. Many fish won't bite a worm or other bait swimming 12-15 knots through the ocean - this isn't a natural mealtime option for them - the bigger fish may (marlins, tuna), but these are very large fish requiring a fishing operation, not a simple sailor fishing for his afternoon cup of chowder. The option then would be to take down the sails, stopping the vessel for a few hours to allow the men to fish (and fish populations tend to be much more plentiful near the coasts, rather than in the shipping lanes of the deep sea), which could take a lot of time and resources and leave the men on the high seas (where weather was their worst enemy) for a longer period of time. Conversely, they could travel nonstop to the nearest port of safety simply by storing food in some of their holds.
I think sailors did not enjoy the months at sea. Stopping to fish so often would not yeild such high results, and there was no guarantee that the fishing would be so successful - meaning the men could starve. I doubt many sailors would have joined with the caveat that they'd have to fish for their own food. Sailors did consume more than simple hard-tack, though this was the light and portable staple, along with alcohol :-)
Even if some sailors did desire to fish for extra tasty bits to make some fish pate for their hard-tack crackers, there are still a few problems. Any moving ship would inevitably cause all the lines cast on either side of the ship to crift toward the same location, toward the ship's stern (rear). So really you could only have one line on either side without the multiple lines getting caught. This hardly makes for a dependable food source for 30, 90, or 160 men. Throw out a net you contest, and the ship is now dragging a lot of weight and moving quite slowly through the water, adding hours or days to the length of the journey. On a crowded ship over the course of several months in crossing the sea, every day spent on the ship needlessly endangers the captain to the risk of mutinee. These men DID NOT want to be on these ships for very long.
Finally I think a lot of these sailors were inately afraid of the sea. We think of them as sturdy seamen, but by and large, until the advent of the professional British navy, they were largely men who had little or no experience on the seas.