You can definitely use Shopify to leverage and support your online courses business. There are apps that also assist with just that. I've gone ahead and compiled several apps I feel will have some great features that you may want for your business:
Locksmith - This will allow you to lock pages and content on your store so that only the customer/affiliate with the credentials to see that page will have access to it.
Refersion: Affiliate Tracking - This is a great app that you can use to power an affiliate program if you plan on partnering with others to market and recommend your site/courses for a commission.
Digital Downloads - This can be used to help with any downloadable content that you offer to customers, they can make the purchase on your store and will then get a link to access that download.
Bold Recurring Membership - This could also be an option as you can offer member access for tiered pricing and courses.
Courses - This is sort of a pre-packaged online course platform solution that allows you to create lessons and sell them online through their app. You will be able to add necessary materials to your products (courses) – text with all necessary formatting, images, embedded videos, etc (in a similar way to Udemy, Coursera, Lynda). In case if webinars format works better for you, you can embed online videos or even live streams into the lesson's contents.
If there is anything else I can help you with, please let me know.
Shopify is implementing the "Shopify Educational Partner", which verify courses and give it a "Shopify Approved" badge. I think it's a good way to promote your course and you might be interested.
If you have any question, please ask in this Q&A topic, I'll be there answering questions regarding store design and technical design.