As Dave says, the 'main air' resonance will effect the notes that are close in pitch to it. If it's exactly on a played pitch, that note can be problematic in a few ways. For one thing, the guitar will tend to suck all the energy out of the string and dump it in the room, so you end up with a note that's twice as powerful for half as long. Since our ears are not good at detecting even fairly large changes in power output, we don't notice that, but the sustain issue sure jumps out.
Believe it or not, a very strong 'main air' resonance can effect intonation. This is because the air pushes on the top, which moves a lot, and the resulting motion of the bridge can cause the string to think it's longer or shorter than it's supposed to be. The notes just below the 'main air' pitch will tend to be a little sharp, and the ones above it a tad flat. When the 'main air' pitch matches a note exactly that pitch can be fuzzy or buzzy, as the string tries to decide how long it really is. If the 'main air' pitch is just above the pitch of the open A string the result is that you tend to leave the string a little slacker than it should be, since it's sounding sharp. The A# is shifted flat by the top motion, though, and since you've already left the string a little slack, it's even flatter. The notes get a little closer to the right pitch as you go up, but they never really catch up to where they're supposed to be. The tuner says the open string is right, but the guitar sounds awful.
BTW, you'll note that I keep reffering to the 'main ar' resonace, rather than the 'Helmholtz'. It's a small point of usage, but meaninful to us acoustics geeks. Technically, you only get a real Helmholtz resonance in a box with rigid walls. Helmholtz himself used glass spheres, and some people say that's the only 'real' Helmholtz resonator. I woudn't go that far, but you do need 'rigid' walls, at least. To see the 'real' Helmholtz resonance on a guitar, you need to do something like what Rossing did, and bury the thing in sand with just the soundhole open. On a guitar the top (at least) interacts with the air pressure changes inside; compared with a rigid box, the Helmholtz-type resonance on a real guitar will be lower in pitch. We generally call this mode the 'main air' resonance.
This gives you a way to 'fudge' things on your pair; you can make the walls of the box with the higher 'main air' pitch a little less rigid. Usually the best way to do this is to reduce the stiffness of the back. Backs are often a lot stiffer and heavier than they need to be, and it won't hurt to shave some wood off a brace or two. The top, OTOH, is under a lot of stress, and it's too easy to take off one shaving too many and compromise the strength. Besides, it's easier to get at back braces....
