Is it OK to train a binary classifier using all the extremely imbalanced data if the majority class is negative?
An imbalance of this magnitude is definitely a problem. Because machine learning algorithms penalize model complexity and try to maximize fit, a model that always predicts "False" will often outperform any other model because it predicts the right outcome for 99% of the data and is super simple!  

What one should do in this case is downsample (http://www.simafore.com/blog/handling-unbalanced-data-machine-learning-models), train the model, and then scale the resulting predictions back up to reflect the imbalance in the original data.