The DialogBody control has a boolean property called AllowUserContextConfig. Set that to true and the 'Run as owner' option should display.
In your activity execute method, you do not need to do anything special for the run as owner to work. Just use the __Context.Web like normal to access the site. When run as owner is selected, that context Web will have the rights of the user that published the workflow instead of the initiator. The infrastructure takes care of this for you.
Hope this helps