I always use the dbx-on.
That's the way the 388's supposed to be used, by design, and that's a way Tascam eeks out a little extra high fidelity out of the 1/4" 8-track format.
The dbx system really quashes tape hiss, and boosts dynamic range, and is amenable to almost all program material. Although hard, percussive piano, and actual percussion like drums are the hardest signals for dbx to track, I've almost never had any audible side-effects from the dbx.
In general, when you turn off dbx, you're inviting audible tape hiss back in, and there's a slight decrease in overall S/N ratio.
On the 388, as well as actual "Portastudios", the dbx is meant to be on all the time. Dbx -on is what seems to give best results, overall, and that's how I recommend using it.
You may find members who don't like dbx, for "pumping" or various other reasons, but I've always used it, and have had basically no problem with it. Dbx definitely quashes tape hiss and boosts S/N, which is an audible improvement over running without dbx.