Plant hormones usually give either Go or Stop signals. Auxins, cytokinins, gibberellins are the examples of the former, and ethylene, abscisic acids are the examples of the later. Aspirin (ASA) is more of the later, it generally signals to stop the growth and put efforts into defenses. This is a bit of simplification, but there are evidences showing that SA will reduce the root growths in some species. I wouldn't use it unless the plants are under stress. But I used to give quite a bit continuously for orchids for a couple months (before I discovered the paper about suppressing root growths), I didn't notice particularly bad or good. It is a short time, but the negative effects must be fairly small. Now I only use it when I receive plants, when I'm deflasking, and when the plant shows a sign of stress/infection.