No, school staff should not serve on a charter school board. Having staff as board members undermines the authority of the CEO. In addition, board members must discuss the CEO’s salary and performance review, as well as compensation for other staff members. It is inappropriate for school staff to be involved in these discussions.
Teachers are very important stakeholders in the school. Seeking regular input from staff is a management function. An effective CEO regularly consults with the staff about policy matters and brings this information back to the board. This has proven to me a more effective strategy than "seats on the board".
There are a few states that require teachers to serve on charter school boards. If this is the case in your state then of course you will need to have a teacher serve on the board, and should be very clear about their roles and responsibilities and set up some structures so that they do not undermine the authority of the CEO.
Updated