This is an area where you will get many differing opinions. An equity line can be a very viable option in starting a business, just realize that you are putting your home at risk if you default on the loan. This can make some individuals very nervous. Though in reality, when you borrow money for a business, you most likely will have to use your personal assets as collateral anyway. The key is making sure you have a well thought-out business plan and model in addition to being able to "handle the worst case scenario" if the business doesn't go as planned. There's always going to be some risk, the goal is to make it a calculated risk that you've devoted much time and energy into the forethought of the business plan and business model to minimize your risk.
The fall in new company starts does not necessarily mean that the American economy is less dynamic than it used to be, or that Americans are opening up new establishments at a lower rate than they did in the late 1970s.