I'm skeptical as to how much difference the air volume between 32's and 37's would make, but tires that are available as 37's tend to be heavy. To some extent that's true of 32's also, but there are some speed-oriented tires that come that narrow. You really need to start by looking at other attributes.
In general, with tires you're always looking at trade-offs. You need to decide what your priorities are. Consider the following:
Price
Durability
Puncture protection
Ride feel
Weight
Traction
My favorite tires for any given ride are the Schwalbe Ultremo R.1. They're very light and have an amazing ride feel. On the other hand, they have poor puncture protection and after only 1500 miles they look like swiss cheese. An almost precisely opposite tire is the Schwalbe Marathon Plus. People on this forum routinely report getting 10,000+ flat free miles from them, but they're heavy like nobody's business and have a relatively harsh ride. That's the kind of trade off you're looking at. As you pay more for a tire, you can get more of the things you want, but there's always a limit.
A tire that feels soft will wear out quicker than a tire that feels hard. You can usually spot this in the technical specs by looking at thread counts. Just like with bed linens, higher thread count feels nicer but isn't as durable and is more prone to punctures. A simple way to make a tire puncture resistant is to put a thick piece of hard rubber in it. This works well, but the tire will be heavy and have a harsh ride. Many punctures can be prevented by a lightweight barrier such as a kevlar belt. This keeps the tire light, but the price goes up. The ride quality also drops a bit. Aside from ride feel, better quality rubber also improves traction.
Skinny tires let you run at a higher pressure, which is good for speed, but if you don't keep the pressure high they become more prone to flats. Wider tires are inherently more resistent to flats, and you can run them at a lower pressure, which is what makes them more comfortable on rough surfaces because the squish of the tire absorbs some of the bumps in the road. It also absorbs some of your energy every time you crank the pedals. Again, trade offs.