If the young fish are of the rarest varieties, they will not be able to remain completely healthy until you have fully researched Fish Environment (Level 3). If their health gets into the red, and you have enough money, you can buy vitamins and drop them into their tank. This will boost their health enough that they'll probably survive into adulthood (at which point you can sell them).

Also make sure that you're careful not to overfeed them. Feeding your fish too much or too often can harm them. If you feed them before they are hungry (about halfway down on their "Fed" meter), they will lose health each time you feed them.