Does having more curvier corners help warping more then smaller curves
In short, yes, it helps a little. Curves provide less surface area per unit volume (a sphere has the least possible surface area relative to the volume of the solid), and that reduces the rate at which the material in that corner will cool relative to elsewhere in the print, and also changes how the material can deform if and when it does cool unevenly. Uneven cooling of fine features, including sharp corners, is what causes warpage, as these areas typically cool faster than others. The curve also helps with bed adhesion, as the more rounded corner is more easily kept down on the bed by the various forces within the object and in the bed adhesion. Sharper corners depend more on the surface area under the corner itself to keep the piece properly stuck to the bed.

However, it's not a panacea. It's not always possible, for one; it's usually an option when CADing a "green field" design for a household item or other standalone product, but if you're making a replacement part for an existing device, or printing a figurine or other detailed model, you usually have to take the curves (and edges) that design gives you. Also, if you radius an outer edge of a hollow shape, but don't radius the inner corner to keep the material thickness constant, you'll end up with similar cooling problems as the apex of that edge cools fastest.

Strategies for avoiding warping are along a couple common lines, but exactly how you implement that strategy depends on the material and on the printer. One overarching strategy is to increase print surface adhesion; the stronger the part sticks to the bed everywhere on its first layer, the harder it will be for corners to lift. Exact techniques depend on the print material, but many of them, like blue painter's tape, Elmer's glue and hairspray, work well for several filament types. Adjusting the printed shape, to print a brim around the shape's base or even a raft under the entire object, are also common anti-warping strategies. The tradeoff is that the stronger the bed adhesion, the harder the part will be to remove from the build surface when complete, which depending on your print bed construction can cause more problems than the warping. 

Blue tape is nice because it's cheap to buy and easy to apply to the bed surface, both of which mean you don't have to worry too much about damaging the tape as you remove the part, just strip the tape and put down another layer. But, it doesn't work for all filament materials; the only thing I've found that really works well for ABS, for instance, is kapton tape, which is significantly more expensive and takes a lot of effort to lay down a wrinkle-free, bubble-free layer over the entire print bed for a large part (especially on my MakerBot, which doesn't have a removable build surface on it, so I'm laying this tape down in cramped quarters within the printer enclosure). A glass surface painted with ABS slurry is an option I've not yet had opportunity to try for ABS printing, but plenty of people swear by it.

Cooling, especially uneven cooling rates, are another major contributor, but again, exactly how you deal with cooling depends on the material. Most plastics, especially PLA, tend to work best when you keep things as cool as possible; the coolest extruder temp that reliably feeds fil, the coolest bed temp that reliably sticks, part cooling fan turned up, and even a standing or box fan blowing through the entire work area to put as much air over the part as possible. What this does is to get the PLA down onto the print surface as a hot "putty", then immediately "freeze" that bead of plastic as a solid as the heat's removed, so the bead can't shrink as it cools more gradually. 

Now, with ABS, this cooling strategy doesn't work, and in fact it's the worst thing you can do to an ABS print. The material is much more elastic than PLA, which is quite brittle, and has no true melting temperature, with a very hot glass transition temp. So, as it's laid down, the extruded beads quickly put the part under elastic tension as it cools. Shrinkage is a fact of life with ABS; the only thing you can do is to control the shrinkage by controlling the part cooling, so the part cools and shrinks evenly. ABS calls for a hot print bed for good adhesion, which will keep the first few layers warmer longer, but higher layers of the print will be further from that heat source, so if there's air moving over or within the work volume, these higher layers will cool more rapidly, at which point it really doesn't matter how good your surface adhesion is (I've seen prints split halfway up the model to relieve the tension by warping). So for ABS, a heat enclosure is pretty much a must. My MakerBot is built that way (in fact it's designed for ABS printing and works better with ABS than with PLA, which is supposed to be the more user-friendly material of the two), but most of your open-gantry RepRap-type designs will need something built around it, usually with a separate heating element to heat the work volume more than the bed is capable of doing.