Summarize this article:

Your bra straps should be doing minimal work to hold up your breasts--a properly fit bra band should provide almost total support. If the band is too loose, you may be tightening your straps to pick up the slack, causing them to gouge into your shoulders.  Your bra band should be snug, but no so tight that you can't slide a finger between the fabric and your skin at the front of your bra.  The band should be parallel to the floor and sit no higher than the bottom of your shoulder blades.  If you measure your bust and find your bra band is the right size but still too loose, then it's simply too old and needs to be replaced. If your cup size is too large, you may be tightening your straps too much to keep the cups from looking pointy or to pull the them flat against your chest.  If your cup size is too small, the straps may not be long enough to accommodate your breasts and be pulled tight, gouging your shoulders. A bra in the correct size will have cups that lie smooth and flat against your chest, without pinching or causing your breasts to overflow from the sides or top.  If the center panel of the bra (the part between the cups) doesn't lie flat against your breastbone, your cups may be too small, forcing you to tighten your straps to a painful degree. Many full-support bras come with wider straps, which distribute weight better than thin straps and provide more comfort. Even with a properly fitted bra, thin straps can sometimes still be painful for women with larger breasts. Try to avoid them when possible. If your bra fits properly but the straps are still causing you discomfort, invest in cushioning designed specifically for bra straps. They fit under or around your straps and should mold against your skin so they are not obvious under your clothes. Some bra straps, especially wider straps, come with build-in cushioning for added comfort.

Summary:
Check that the bra band is the right size. Make sure your cup size is correct. Wear a bra with wider straps. Buy a cushion to wear under your straps.