No, it won’t. Most colleges require around 120 credits, which for an average of 3 credits per class works out to around 40 courses. For example, check out Stanford’s major - there are six core courses, a speculation of 4-5 classes, and then 2-4 elective classes. That’s around 15 classes at the high end, which is about average for a major at most colleges. There also cognate classes in math and physics - so CS majors at Stanford need to take two math electives and two physics electives. So let’s just round up to 20 courses.
That leaves OP with 20 other classes that they can take in different fields. Some of those will be distributional requirements, but you can usually pick your own classes within a certain division (so if you have to take a humanities class, that class can be a literature, history, or philosophy class) Stanford also has joint majors with computer science - so there you can pair CS with comparative literature, Iberian and Latin American cultures, Spanish, classics, English, French, German studies, Italian, history, linguistics, music, philosophy, and Slavic languages and literatures.
I checked out MIT’s EECS requirements - one of the most rigorous programs in the country - and they also have 15 courses. There are probably some mathematical cognate courses I am missing that bring it up to around 17-20 depending on how much math and physics a student comes in with already. In fact, like most other universities in the world, MIT actually requires you to take courses in other things - including 8 classes in the humanities, arts, and social sciences.
That’s not to say that you won’t spend a lot of time on CS classes and projects, but every major spends a lot of time on the projects in their majors. English majors may spend a whole weekend finishing up a paper; sociology majors may run around trying to get together participants for an independent study or thesis project. I’m not even saying that CS majors aren’t somewhat more rigorous than other majors (although it really depends on the school and the departments themselves). But if you go to a traditional liberal arts college or university (which most places - even MIT - are) then sure, of course you’ll be able to explore other subjects.
Whether or not you need to get a BA will depend on the school. Some schools don’t offer you a choice - everyone gets either a BA or a BS, and the requirements are the same. At schools where you do have a choice, the BA is usually the one with fewer requirements, and as was mentioned your employers won’t notice or care whether you have the BA or the BS. What they care about is what you can do.