What is the best way to open a Japanese branch of a US company?

Your options include;
1) Set up a subsidiary (e.g. K.K.:similar to C-Corp, or G.K. similar to LLC) in Japan
2) Set up a branch office of the Delaware C-Corp and get registered in the Commercial Registry in Japan
3) Just send or hire your staffs in Japan and do some business activities without making revenues and profits.

These options vary in cost, social credit, taxation, legal responsibility, visa issues etc. You should decide based on what kind of activities you do and what are required from the viewpoints of your customers or local partners.

As a recent trend, increasing numbers of American businesses in Japan use the form of G.K. (similar to LLC), such as Apple, Cisco, Seiyu (Walmart) because it is more simple than K.K. (similar to C-Corp) in the form of internal organization structure (e.g. No need for internal auditor in the board of directors).