What can I do about leaky breasts in the third trimester?

See answer. This can be an issue and is common. Discuss with your gyn, and you may need to invest in some good maternity bras with padding to prevent the leakage from being noticeable through your clothing.
Wear a tight bra. Wearing a bra a size too small may decrease the leaking. Otherwise, nursing pads in the bra cup can at least decrease the mess.

Related Questions

Is it normal to have leaky breasts in the third trimester?

Yes. Many women produce colostrum in the second half of pregnancy. It is totally normal.
Yes. The hormonal changes that occur during pregnancy will prepare the body to produce milk and often can cause leakage late in the pregnancy.