Yes, you can start many businesses with no money down, so many that it's impossible to even start listing. Most of them are services. Think of businesses like graphic arts, landscaping, gardening, carpentry, consulting, business plan writing, painting, home car, bookkeeping, so many other kinds of businesses ... what you need is customers, not a down payment. What can you do that people or businesses will pay you to do?
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.