The world is suffering from infertility, and the founders of Gilead use the example of godly intervention due to uholy behaviors in modern society, especially women who have denied their god given ability to bear children by commiting sacrilegious acts such as using birth control and having pregnancies aborted. Excessive pollution causing climate change is also indicated as an underlying factor for the drastic change in fertility.
Little is known how other countries cope with this, but in the tv show, it is strongly implied that other countries are attempting to obtain Handmaids directly from Gilead via trade agreements, or at least learn from the culture that has created them to try to enforce some sort of similar child-bearing role on their own fertile population. Offred pleas with Ambassador Castillo, the mexican abassador, begging her to take heed of her words, saying her and the other handmaids have not chosen this life and are being held against their will to be used in sexual servitude. However, Ambassador Castillo responds telling Offred "I am from Xipica. There hasn't been a child born alive in Xipica in six years. My country is dying." Later in the same episode, when June is talking with her friend and close confidant, fellow handmaid Alma, the conversation leads to the fact that Alma has overheard conversations from her commander indicating Gilead was making trade agreements including Handmaids as products for export.
Also in the tv show, Moira was a surrogate for non birth parent in exchange for money so they may be using the eggs and sperms from fertile people through the technological means to procreate more babies. Gilead uses a different method to procreate in the most natural way at the cost of the fertile women’s freedom by forcing them to become Handmaids if they are sinful in the eye of Gileads laws.