First to clear up a commonly confused point, not all of those barrels are Damascus, some are laminated, some are stub, etc., and there are quite a few varients of multiple metal barrels.

Barrels of this type give up nothing to solid steel barrels in terms of strength.

The bad reputation stems from the following problems:

Some of these barrels were simply cheap junk and as such would still be cheap junk if made out of solid steel.

Many were designed for BP loads. They simply were not built to take modern high pressure loads (This applies mainly to shot guns.) The same problem would still exist if they were solid steel. This problem is aggrevated by the use of longer than chamber cartridges in shot guns.

While these barrels may not rust any faster than a solid barrel, it's possible for the rust to be hidden. Probably applies more to cheap barrels.

Given a good quality barrel in good condition with the proper load there is no reason these barrels cannot be used safely.

The BIG caveat is they do require understanding to avoid falling into any of the pot holes above.