All wood stains, whether they are oil based or water based, do not increase the susceptibility to rot or absorption of water. If they do anything, they will add to the longevity of the wood that is in contact with the soil.
If it were me, when I stain the outside of the finished bed and if I had stain leftover, I would paint the inside, too. It is generally better to use up products as there were intended, rather than throw them away in the trash. However, if I plan to grow edible plants in the bed, I wouldn't paint the inside.