Well, two that would cause a lot of concern in any period that might be related to the play would be a lack of most reliable contraception and a great risk of dying while giving birth. A third that would be related would be sexually transmitted diseases. The condom mainly started to be used in the seventeenth century, so if R & J had sex, they would risk pregnancy and maybe disease, and she might risk death.
Of course, a more common risk and one harder to avoid would be the plague or Black Death. It moved through heavily populated areas, so much so that London closed the theaters in 1608 to try to fight it (including closing the Globe).
Even more common? Poor diet, poor dental care, and exposure to the elements (no good heating, etc.).