You are mostly the same person, and you would still be conscious because consciousness is in your genes too. Materialistically speaking. Personally my beliefs lie elsewhere, but I think these two viewpoints capture the basic dichotomy of approaches to this idea.
Well... I guess alterations to the DNA now wouldn't do anything at all, because coming to think - alternations in DNA shouldn't remove you of your memories, so you will still know exactly who you are, who you have been, and etc.

But if the alteration was to come before birth - do I still end up here? Am I 're-assigned' to another physical body, starting with the slightly altered one - or so I never experience reality period. Over the course of human existence there will only be a finite number of experiences.

May be a bit confusing to some, but a few already seem to know where I'm going.