I know that if my shared links generate too much traffic. It will be banned. I'm using Dropbox (Basic 50GB) as a server for my application to run which i'm sure will generate a lot of traffic. But, can i overcome this by using the Dropbox API.
No, the API is subject to the same limits, you can't avoid the restrictions in that way.
Richard's correct, any links retrieved via the API count toward an account's link limits, just like any other links. If the content is just being accessed by your app directly though, you should use the file downloading API calls, which don't return links and thus aren't subject to those limits:
v1: https://www.dropbox.com/developers-v1/core/docs#files-GETv2: https://www.dropbox.com/developers/documentation/http/documentation#files-download