wider tires are usually more forgiving for stuff like, curbs, potholes, rocks, etc. they are usually lower PSI, which means they have more "give" in them. mainly a wider tire means the weight is distributed over a bigger area, so if you hit something that could puncture your tire, more weight is on that given point with a thinner tire.

just going to 32 or 35 won't do much. get a puncture resistant tire, and (this is what i mainly do) just avoid potholes, gravel, anything that would puncture a tire. i do this (running 26x1.0 tires, 70-80psi) and i've been flat free for...a while now.