If the cabinet is typical chipboard panels which either have melamine or a foil finish, once they have swollen there is not much you can do to fix it. If you are looking to prevent further damage then you need to find the cause of damp getting to the material. In a bathroom there are likely two causes condensation or direct water contact. For the first, if you have someone who likes steaming hot baths but keeps all the windows shut then the lack of ventilation will likely be the cause of condensation, opening a window, a better fan are the solutions there. If the cause is direct water contact on edges that have edging strip and will be visible then a plastic angle liberally glued on should fix the seal without looking too bad. You say the bottom edges are where the swelling is taking place if these are simply cut edges without edging strip then a liberal smear of silicone along the edge will make an effective seal, do any other similar edges.
Richard