You'd want to crush it much more finely for it to have a chance to actually capture anything while in the air, requiring more energy input and causing breathing hazards due to the fine particulates, not to mention staining everything it rains on with the fine basalt.

You could release it extremely high such that it stays suspended a bit longer, but then you'd waste even more energy getting it up there.

there will be very very little dust fine enough to become suspended in air, without running it through ball mills, which will be stupidly expensive. "crusher dust" is the label given to the fines from rock crushers, that is suitable for use on trsditional "gravel paths" in parks. Basalt is actualy quite friable, and crumbles fairly easily......depending on the type...., some would be more trouble than it's worth.....other types are very crumbly and adding the rock to agricultural soils, has the added benifit of incresing soil fertility, while absorbing CO². So the initial payback from increased production and quality of food, ofsets a portion of the costs, prrhaps all of the costs in situations where the right types of rocks are located near good agricultural candidate areas. The other agricultural carbon capture method is tera pretta or biochar, where organic waste is chared and mixed into soil, where it persists for many millenia, and provides a media for soil biota to live in, and reacts with varios harmfull contaminents to sequester and neutralise through chemical and biological reactions. The impilimentation of any carbon capture method is going to be dependent on local conditions, that will vary quite considerably, ie: what makes sense and is actualy benificial in one area, could easily be a net carbon cost, somewhere else. The main variables will be crushing ,trucking, and spreading costs, and then I think,actual soil ph and chemistry, as certain soil types would prevent any reaction with the rock, and atmospheric carbon. The whole thing, bieng a thing, and a deap rabbit hole wanting dedicated local expertise and knowledge. A personal observation of "carbonate beds" leads me to believe that under certain conditions, the amount of material produced is truely huge.