Blender does not have any functionality to use direct weight painting on meshes without faces. Nor can you effectively weight paint a mesh that is composed of geometry dynamically generated with geometry nodes-- you need to apply the modifier.
You can still assign vertices to weight groups, using the properties viewport's Assign button or with your vertex groups specials menu (ctrl g in edit mode for me.)
When I work on string meshes that need weights that I don't want to make via brute assignment (usually, where I want to use a gradient tool to set the weights), I extrude the mesh in some direction, assign the extruded vertices to a new vertex group, and create a mask modifier to delete those new vertices dynamically. Then, whenever I disable the mask, I can paint on the extruded faces, but the faces will be deleted automatically by the modifier stack on re-enabling the modifier (or on rendering.) This is only a few operations-- extrude, assign, set up the mask modifier-- and so is easy enough for me. Whenever I have decided that I don't need to edit weights any further, I can apply the mask modifier to delete those vertices and faces permanently.