Wet soils after alfalfa cutting can encourage weeds and grasses like foxtail, crabgrass, pigweed, and others to grow quickly.
The best management method is to keep alfalfa thick and thrifty so it will compete aggressively with invading weeds. This begins with ensuring thick initial stands and good soil fertility and harvesting alfalfa only after it begins to bloom or when new shoots appear at the base of the plants. This will help alfalfa regrow rapidly so weeds don't get a foothold.
Unfortunately, this method is easier said than done, and forage quality will be lower since harvest occurs after bloom begins.
Herbicides are another option. Roundup works great for Roundup Ready varieties. For conventional alfalfa varieties, two chemicals that control annual grasses are Select Max and Poast Plus. These herbicides work well on seedling grasses that are less than 4 inches tall and are well tolerated by alfalfa; however, neither herbicide has any soil residual activity so good plant coverage is necessary and repeat sprayings may be necessary if new grasses emerge. For broadleaf weeds, Raptor and Pursuit are your best choices. They, too, need to be applied before weeds are 4 inches tall. Fortunately, both Raptor and Pursuit have some residual activity and can be applied a little early and still provide control of many later emerging weeds. They will, however, also set back your alfalfa just a little bit.