I'm certainly no expert at this. However, if it were mine, I would sand it first and see what happens. It appears from the photo that part of the finish has dissolved and part hasn't. Sanding it would take off the high spots and leave the low ones. We know it's a veneer. With the open grain, it appears to be oak, probably white oak. Those edges have to be replaced with new pieces of veneer or filled in with some kind of filler that will blend in. If it's only on the edges and minimal, you can probably fudge it with a good wood putty (depends on how picky you are).
Rockler is one place that has what you need to do this. If you can go to their store, they offer good advice as well.
Start with fairly fine sandpaper; I'd use an electric sander. Go easy and sand with the grain. Use a paint brush to brush off the sawdust so you can see how it's going as you do it. I would think stripping would be a last resort, depending on how well the veneer is adhered.
A quote from my Uncle Edwin: "Elmer's glue and sawdust has saved many a carpenter's reputation."
Janey, you'd have to really want that table to tackle that job. It's easier to find another one that isn't in such bad shape. Either sanding or stripping is the only thing that will help. The question is whether it's a good enough cabinet to be worth the effort.
bkay