That is usually caused by a rise which is too short for your proportions - especially in this era of low-cut. Look at my site and learn the various proportions which are available. Find something which would seem to work better for you. It sounds like the 252/846 Classic Boxer style might fill the bill and you can see it here: http://www.customshirt1.com/ZM252_01Boxers.htm
Then check some of the larger underwear sites (HisRoom, UnderU, SimplyMan) and see if you can find a brand in your price range with a model which looks like it has a longer rise. The other suggestion, as noted above, would be a fitted model containing at least a portion of Lycra such as the 700/1343 Closed Fly Pant: http://www.customshirt1.com/ZM700_01All.htm
A wide range of manufacturers have some sort of model similar to this one. Of course, there are always our bespoke boxers @ $165 ea.
Thanks guys. I'll look into different-fabric ones for sure. I'll try different sizes as well, but I think I naturally have a lower rise. My underwear tends to often show "bunching" between waist band and crotch, and many pants are "crotchy" on me. (My body proportions tend toward longer limbs and a shorter torso, which I've heard would give me a shorter rise for my height, but I could be wrong.) I forgot to mention that my BBs mostly ride up on my left leg; they stay put on my right one. Not sure what this means. Anyway, thanks again.