Some of the answers seem to be saying that you want the body to resonate at the frequency of the sound so as to produce the maximum amplitude of sound. That's not quite right. The graph below shows a measurement of the resonance curve of a 1713 Stradivarius violin (redrawn by me from a figure by Carleen Hutchins). There are a number of different resonance peaks, some strong and some weak; the ones near 200 and 400 Hz are vibrations of the wood, and the one near 300 Hz is a resonance of the air moving in and out through the f-holes. The white lines show the fundamental frequencies of the four strings.
So you can see that there certainly are peaks, which indicate resonances, but they're rather narrow and there are a lot of them. From what little I understand of violin acoustics, the effect on the sound is complicated, and has to do with the way the different harmonics coincide with the many different resonances. When you play with vibrato, you're sliding the harmonics back and forth over these resonance peaks. In any case, this is definitely not a situation where the fundamental frequency of the note simply matches the resonance frequency of the body and/or air cavity.
The OP specifically asks:
Why do lower pitched string instruments have a larger body?
In other words, why can't we have a double bass with a soundboard the size of a violin? The answer really has less to do with resonance of the body than with the size of the body in relation to the size of the sound waves. Each of the resonance peaks in the graph above corresponds to some pattern of vibration of the soundboard. Below are some diagrams from WP of the patterns of vibration of a guitar's soundboard.
The crucial point here is that in all of these patterns, there are some areas that are rising while others are falling. That is, different parts of the soundboard have different phases. Now the lowest note you can play on a double bass is 41 Hz, which corresponds to a wavelength of 800 cm. Suppose you tried to produce this sound using a soundboard the size of a violin, where the different vibrating patches were only 5-10 cm in size. Then for any patch that was vibrating outward, trying to produce an overpressure in the air, there would be another patch right nearby that was simultaneously vibrating inward, trying to produce an underpressure. They not cooperating. They will come very close to canceling out.
Let's refer to the patch that's trying to produce an overpressure as +, and the other as -. Ideally you would like the the + patch and the - patch to be 400 cm apart. This would be half a wavelength, and then the + and - would actually be cooperating on producing the same 800 cm sine wave: one would be producing the crest, while the other produced the trough.
In reality, the size of a double bass's soundboard is somewhere in between these two extremes. Because it's pretty small compared to most of the wavelengths it plays, there is a lot of cancellation going on, and not much cooperation. However, the cancellation is not perfect, so you do get some sound. The instrument could be made louder by making it bigger, but that would be impractical.