Mowing ragweed can keep it from flowering and setting seed, and as common ragweed (Ambrosia artemisiifolia) is an annual, that should hopefully lead to significantly less ragweed in future years.
Birds also love the seeds! Goldenrod (Solidago ... The true culprit that causes allergies in the autumn is ragweed (Ambrosia artemisiifolia), which blooms around the same time as some varieties ...