I have an anthropology degree (switched from CS), and it hasn't hindered me. You'll be fine either way. I'd make sure to work on some interesting side projects

I have an anthropology degree (switched from CS), and it hasn't hindered me. You'll be fine either way. I'd make sure to work on some interesting side projects and contribute to open source projects. Lack of experience is more of a hurdle, and showing that you've been working on things outside of school can help you get past it. I can't speak to grad school.