You do not need any particular qualifications to be an illustrator, although it may be useful. To be an illustrator, you need to have a large interest in art, and an interest in other illustrators' work.
Taking a course in illustration will help you to understand what type of projects illustrators undertake, and how to organise and present your portfolio of work to potential clients. A degree in illustration would be useful, as you will receive lectures from established illustrators and portfolio consultants that will talk you through the processes of illustration.
You will cover all areas of illustration in your course, which will be useful as it will help you decide what type of illustration suits you best. During your university course, you will participate in critiques, which will allow you to view the way that other students have tackled the modules you have been set, and you will be able to share your work and techniques with others.
If you don't wish to do a degree in illustration, then perhaps a degree in graphic design or fine art may be useful, as both of these intermingle with illustration.