The question is pretty much the title. To start off, I'm not really worried about my record in pure mathematics classes/research. However, solely for the purpose of having a career backup I've also picked up a second major in applied math and several CS courses. I generally have little passion for these classes and ended up with a fair share of B's. My overall GPA is still fairly high but I was wondering if these grades would hurt my application to pure math programs and if I should address this in my personal statement.
The GPA is certainly one of the competitive aspects of an application for a graduate program in mathematics (and presumably, in most other academic fields), so anything which brings down your GPA hurts your application some positive amount. Lackluster grades in applied math and CS are probably going to hurt more than lackluster grades in history and cinema studies.
As is often the case with admissions advice, it's hard to be more precise because the individual admissions committee gets to make up its own mind on how strongly to weight everything. (And this may not be a bad thing: this is what allows one school of a certain caliber to accept your application while another school of the exact same caliber rejects it.)
For this issue, the structure of the department and university may play a role. Namely, for undergraduate programs at US colleges and universities (your profile lists you as living in the United States) it is relatively rare for pure math and applied math to be truly separate majors. In my department for instance applied math is an "area of emphasis" within the math major. Yet more commonly, there's just one major and students show where they fall on the spectrum by their choice of classes. If the graduate program to which you're applying is really just a pure math department (again, relatively rare at American universities, though my own graduate program was like this), then they may (perhaps) be used to ignoring or discounting applied math courses. If there's just one math department then admissions faculty who look at your transcript are probably going to view the applied math classes as math classes, and the amount that you will get dinged for your performance will probably be non-negligible.
Now comes the time to say that I find your overall approach curious. You should not major in a subject for which you have "little passion". If you know you are not interested in something then the obvious -- a bit narrow-minded, but strategically sound -- plan is to take the minimal number of courses required in that subject but make sure that you do well in them. (Most often required undergraduate level courses at US universities are not too challenging for the type of students who are going on to graduate study.) You don't like applied math or CS but you might, as a consolation prize, want a career in them? This doesn't really scan. I think it is much better to pursue serious career goals one at a time: if you don't get into graduate school for pure math you can still get a real world job with a major in pure math, and if at that time you discover that necessity has given birth to interest in these other fields: so be it, you can learn more about them then.
You ask whether you should address this in your personal statement. Perhaps, but since I find your explanation a bit strange I'm not sure what you should write. Consider providing less pointed information: e.g. you were considering a technical / industrial career until you discovered the depth of your interest in pure mathematics....Yeah, I find it tough to put a good spin on your situation that is faithful to the information given. So a brief "sorry, these grades are anomalous; what I really like is this and notice how stellar my performance is there" may be the way to go.
B's in the major you are not planning to pursue in grad school are not a disaster.
Sure, in grad school you don't want to have multiple B's in your transcript. But we're talking about your undergrad transcript here. B's are allowed.
Your strength in pure math will be apparent in your pure math grades, your math GRE, your undergrad research experience if you have any (this item is optional, so if you don't have it, don't fret), and your essay. I recommend that you speak in your essay about what really gets you fired up, in positive terms. (Leave out anything about how if you never have to write or debug another large piece of code again it will be too soon.)
There are plenty of schools where pure math and applied math are in separate departments. To be on the safe side, make sure at least one of your applications is to such a school.
And tell your parents (who of course want the best for you, but perhaps are not trained mathematicians), real men and women do pure math! (I'm saying that out of support for what you like. I personally like applied math best -- but everybody's different!)