As mentioned in the Comments, we could do with clearer photos to be sure what's going on here, but there are some generalities with commercial speakers that probably cover this sufficiently.
Speaker boxes are virtually always made from some type of manmade board material. So plywood earlier on, followed by chipboard (particleboard) when it became available and then MDF after. The latter two often had a real wood veneer glued to their outside faces, which despite the solid black colour might be what you have here1.
Simply glueing the separated veneer back down is doable, but no sort of proper fix. Regardless of whether this is ply, chipboard or MDF a water-swelled edge can be recompressed but it's generally not that simple to hide the damage completely, but regardless I think effectively impossible to do in situ2. Plus you're probably not set up to do it anyway.
Since I doubt it's financially viable to get a pro to look at this for you (and wouldn't come with any absolute guarantee of success anyway) I think you're left with two options. Three if you can just live with it :-)
- Replacement speaker, or just its box, from a secondhand set of the same speakers. I presume from what I've seen online the 701s were popular enough that this might not be too difficult in some parts of the world.
- New or new-to-you speakers naturally.
- As it seems this doesn't affect the sound output it's really only an aesthetics thing at the end of the day. Perhaps you can position the speakers so you can't easily see this side of this speaker and leave it at that.
1 Could also be textured vinyl or impregnated paper.
2 It requires substantial clamping force from strong clamps and/or a vice, but really requires access to both sides of the affected board. So the box has to be taken apart which I think rules this out.