Yes. Having a passive crossover does not preclude one from doing the same thing with DSP somewhere else in the system. People do it all the time.
Most passive crossovers aren't really doing anything there--you're just stuck with the natural response of that driver in that particular box. My default starting point for a passive speaker is to put a high pass at or just below the tuning frequency of the port (with ported speakers) to protect the driver, then boost that thing until it's flat.
A passive crossover in the system
does preclude you from doing other things you can do with active speakers (like making adjustments to the crossover itself, obviously) but getting a good bass response isn't one of them. Too many here seem to think there's some law against using DSP with passive speakers and are making a comparison of "No DSP vs DSP." IMHO,
everybody should be using DSP--even those with passive speakers. It would be more informative for people making comparisons to limit those comparisons to the things that can only be done with active speakers (steep crossovers, etc) instead of assuming passive speakers won't have any DSP available at all.
Back to the subject of this thread--something I haven't seen mentioned is building codes. There are many places here and around the world where you simply can't have high powered amplifiers built into your walls. This is one of the big reasons pretty much all in-wall speakers--even really high end active ones from Grimani, Pro Audio Technology, etc--use external amps. Even the subs. And yes, running speaker wire through the walls to 15-20 speakers is much easier than a low level signal and having to worry about providing power to each of them.