Do reading glasses make your vision worse?

Not at all. Glasses do not worsen the eye's vision. They serve to improve the vision by supplementing the prescription that your eyes need to see sharply. Reading glasses are magnifying lenses that improve the focus of eyes that can no longer focus well on their own. Our ability to focus up close decreases naturally as we age; glasses simply correct for this and don't cause it.
No. Wearing glasses, whether they are the correct prescription for your eyes will never harm your eyes. All glasses do is bend light and so at worst, wearing glasses will focus the light on a different part of the eye than is required for clear vision (if they are not your correct prescription) and at best will help you see better. Wearing glasses won't make you need them more in the future.