What @mijostyn said, though I disagree with some of the prioritizations mentioned here.
With subs it’s mainly about capacity, design principle and implementation. Through this structural integrity of the enclosure and overall build quality should be "sufficient," but personally I find the need to make them inert/heavy in the extreme to be unnecessary. I’m not saying rigid, heavy cabs don’t make a difference, but to which degree and at what cost (in more than one respect)? Some may find capacity and implementation the most important, others implementation mostly, and others again (like @mijostyn) stress the importance of enclosure inertness (among other things). There are different ways to attain prowess augmenting the mains in the lower octaves.
Capacity, i.e. sheer displacement area and also sensitivity is very important in my book. All things being equal, the less those cones move the lower the distortion, and the less power needed for a given SPL the more headroom. Headroom in the lowest frequencies is paramount (where prodigious amounts of energy can be released), because more of it equates into lower distortion and a cleaner, more effortless reproduction. To boot ample displacement gives you that important physical feel and power of music - vastly overlooked, I find.
Design principle matters. Balanced force approach has been mentioned. By far most subs today use direct radiating drivers in sealed cabs, because this way they can be made as small as possible (and the cabs more easily inert) and retain extension. It’s is the most inefficient approach though, but in multiples this can be somewhat ameliorated. Still, sealed designs have max. cone movement at the tune (contrary to vented cabs and others), and moreover the exposed, direct radiating driver is prone to emit mechanical noise - not least when working harder, which smaller drives in sealed cabs and limited numbers do. Mechanical driver noise = distortion. I prefer large, efficient designs with partially or completely hidden drivers in either horn and/or bandpass variations acting as force multipliers, and with pro drivers no less than 15" in diameter. These designs also bring "inherent" bracing to their enclosures due to horn paths and other design innards, and build in plywood not least are structurally very sound.
Implementation has been covered already. I would also stress the importance of at least two subs, stereo coupling, symmetrical-to-the-mains placement and, preferably, a cross-over no lower than 80Hz. I’m aware this usually involves the need for high-passing the main speakers, and that quite a few audiophiles are against this. Such things would be more easily demonstrated with actual demos to highlight the potential advantages (depending on the ears (and preconceptions) of those who’re listening) of such a configuration, also to narrow down the specific setup context in which the high-passing of the mains has been done - which of course matters a lot to the outcome and to prevent unnecessary generalizations either for or against HP’ing of mains.
So, it’s about choosing the designs that accommodates the above, I find, and this not least calls for the need to include DIY solutions. I’d disregard brands if it means making subs very expensive to get some minimum of physical requirements. In other words, hugely expensive subs from the likes of Magico and Wilson Audio are a waste of money if you ask me, although I’m sure others may disagree. This is below the Schroeder frequency we’re talking about, resonators meant to move air efficiently, cleanly, effortlessly and acoustically well implemented. Trying to make subs into some dubious, "sophisticated" affair akin to selling the idea of expensive, single item (well, two for stereo) small 2-way standmounts is severely sidetracked marketing B.S IMHO.