For health reasons, they're not needed. For comfort/exercising viability, it's going to depend on the person and the exercise. Most smaller breasted women aren't going to need a proper sports bra. Exceptions are women who are nursing, trans women on HRT, and adolescents whose breasts are still growing. For those populations, it's normal for any kind of jiggling to be painful.

Most large breasted women do prefer/need a sports bra to prevent things like skin chafing and grossness that can result from prolonged skin to skin contact + sweat.