Loose Veneer, best Fix?

If only a little is coming off or fraying, you could first try an iron (set to low) and run it over the speaker with a few sheets of paper between the iron and the speaker. This may melt the glue an cause it to stick. This may be your best bet since the glue may be weakening further back. You would still have to use clamps to apply even pressure along the repair. F-clamps with plastic protectors work best. If the repair area is longer than the clamp footprint, use cauls made out of 1x lumber with the clamps. If this doesn't work, then yellow carpenters glue and clamps will definetly do it.

Rja:
If you can find the adhesive type used in manufacturing you are closer to the best possible solution (they might also suggest adhesive type proper for the repair). There can be several types of adhesives used so if you can trim down better.

Definitely if you have the veneer coming loose the faster you fix the better in the long run since larger areas are far more difficult to work and veneer accidental breaking during fixing or use might ruin your speakers not only their looks but also their resale value.

Contact adhesives that have different setting times and might or might not allow relocation so be careful. If you are dealing with small areas the provided bottle tip might not be as accurate as you might need. READ THE ADHESIVE MANUFACTURER RECOMMENDED APPLICATION PROCEDURES AND WORKING TECHNIQUES.
Normally the a containers are also designed for application as well but not be suitable for your particular job, they might designed with large areas in mind so better get some Q tips. Make some trials to learn how the adhesive handles i.e. how fast is sets (this is affected by room temperature, relative humidity so better do some trials), does it streak when you take it from the container? if so, you have to be more careful since it will tend to go outside your work area when you retrieve the Qtip after application. Better to use some paper and quality masking tape lightly applied to avoid a messy looking work if any mishap occurs. Apply the material paying special attention to edges not excesively (adhesion is a surface phenomena and excess material is detrimental to the repair success) to both surfaces to be joined ( need to be free of dust!!) keep them separated (paper clip bent as needed or some other material will do) while the solvent or water evaporates.
Then be sure of this step since normally it is not reversible and how you join the surfaces is how will they remain so take your time, think how you'll do it and go for it!!
Apply pressure evenly avoiding marring the surface and keep that way 'til the adhesive "sets" (see manufacturer recommendation) give it some extra time. Remove pressure application means masking tape and paper.
Sorry if post is lengthy. Good luck.
Luis

Once you find the right cement, and procure a set of clamps (Which are an absolute MUST!!!) get a sheet of wax paper. When the glue is in place, and before the clamps are tightened down put the wax paper where the glue has been placed to prevent the clamps from being glued to the cabinet. It would also be advisable to put something between the clamps and the wood cabinet so the tight clamps don't damage the wood.

In tight spots, wooden toothpicks make nice glue applicators. They're also useful for repairing stripped screw holes.

Elmers glue is fantastic stuff, but with the old adhesive residue it will not bond well. Try to use an adhesive intended for veneering.