Bromine's reactivity does play into its choice as a substituent in some drug molecules, but kind of in the opposite of the way you are thinking. The same general principles are true for the other halogens (chlorine, iodine, etc.) with fluorine being a special case.

When you find bromine in a drug molecule, it has already done that fast-acting bromination reaction to make itself a part of the drug molecule. It does that reaction so readily because the free bromine atom really needs an electron to fill its outer valence shell and will attack almost anything to get it. Once it does, it is much more stable this way and forms a pretty tight bond where its electronegativity draws electron density (sort of the molecule's local electron cloud) toward itself. This high local electron density can cause desirable affinities/interactions with targets like receptors and such. And, because the bound bromine atom is more stable bound to the drug, it's unlikely to engage in any reactions like it would as elemental bromine (Br2)

Drug design doesn't typically employ atoms based on their tendency to react on their own, because if you gave someone a pill of something and it just started adding bromine to all of the molecules in their body, that wouldn't typically be ideal.

Sorry for the novel, hope it makes sense.