My boyfriend can't get over my past. Do I stick around?

You just need to somehow explain to him if you haven't already that you were a different person before you met him and that you are ashamed of what you did. He can't change what you did because he wasn't around at that time so he needs to let go of it if he truly wants it to work. The fact that he always puts you down isn't a good factor for a healthy relationship either.