That is not true. You can be on H1B and not work for a business as an employee but can own a business. Making money on ads on a blog is perfectly legal.
It's a tricky hole to thread: if the blog is being run as a business it could violate the terms of the H1B. On the other hand, a blog that is run as a hobby and that makes incidental/minimal income should not violate the H1B. Generally, the difference between the two is that a business blog is intended and managed to maximize revenue (even if it actually does not make money), while a hobby blog is not intended to make money(even if it actually does).
