If you've been making salad in the bowl, so that oily dressings have gotten worked into the cracks, then it may be almost impossible to fix this. I would just plan on getting a new bowl if/when this one breaks.

I tend to agree. It's questionable if there is a solution that will permanently fix the cracks, especially if the cracks go all the way through the wood. I suppose you could try forcing a waterproof wood glue or epoxy into the cracks, but there is no guarantee it will adhere well, or that the cracks won't break the bond if they enlarge. Wood, even old wood, is something that continues to change over time. Maybe it's time to use this as a display bowl rather than a working bowl...