Does having a minor in Web Design increase my chances of a company hiring me (with a Computer Science Bachelors)?

The degree will get you an internship or entry level job with persistence, initially.

After Job one for 1+ year or so you've cemented your value proposition and should only worry about how specialized your skills are to match new positions.

Years of experience are an HR abstraction to estimate the point at which the average professional achieves specialization competence. Work harder than they think you would. Use new tech. Break things and move fast, think about how to do it better in the next project.

Design will matter at a startup. Startups are better but riskier and pay less short term. Harder work, more hats, work for a startup when you graduate and make some basic stuff in like, AWS like a website in your web language(s) of preference (if you want to embed an interactive application) until that time.

Compliment the schooling with hands-on projects.

If you can get an internship where you can exercise web design, go for it.

Otherwise, create a portfolio to show folks.

In many cases, actions speak louder than words.

I used to be computer science major as well, but throughout one year learning it, I just felt more stress. So I ended up changing my major to visual design with emphasis on Web Design and Motion Graphics.

But as Terri said, to better your chances of getting a job after graduation. Look for internships, part-time jobs on campus, or assistant jobs anywhere. They might unpaid or paid, it really depends where you look and which place you stay.

The real question is what do you want to do with your future? Do you want to go into Graphic Design or do you want to work in computer science? To better your chances for securing post graduation work you should look for internships within the area of work that you are interested in. Look for work on your campus to start, paid or unpaid. Then look in your community for internships. Also as a Com Sci major you might also look into earning some certifications prior to graduation. Experience is what employers are looking for. Get certified, and get some experience. Spend your time wisely and in situations that will augment your career goals. I hope this helps.