If these shoots are coming from the base of the plant, they probably aren't the same variety as the weeping top part of your Japanese maple. Most named varieties, and especially weeping ones, are grafted. Grafted plants are two different varieties; one variety grows on the roots of the other. In the case of a weeping plant, an upright growing cultivar is used for the root system and trunk of the tree, and the weeping stems are grafted on top. Shoots sprouting from the base of the plant develop from the root variety. Your best option is to cut off these shoots at ground level and dispose of them. Although you could try rooting softwood cuttings, chances of success are slim unless you use rooting hormone and have a mist propagation system.