While there isn't a limit specifically on the number of upload or download calls in Basic accounts, the Dropbox API does have a general rate limiting system that limits how many API calls overall can be made in a given period of time.
We don't have any specific numbers documented for that. It is only designed to prevent abuse though, and is accordingly relatively generous. Further, the limits operate on a per-user basis. That being the case, you generally don't need to worry about hitting it in normal use.
Also note that not all 429s and 503s indicate rate limiting, but in any case that you get a 429 or 503 the best practice is to retry the request, respecting the Retry-After header if given in the response, or using an exponential back-off, if not.
