No, a cell phone booster does not require an internet connection to work.
A cell phone signal booster does not require an Internet connection to function. It only requires some existing outside cell signal to work, which it amplifies and then broadcasts to the inside area of the home, office, or vehicle that needs better signal.
A network extender, which is a device sold by the cell carriers, does require a high-speed Internet connection to work. It creates a small cell signal in your home and then routes all cell phone calls and data over your high-speed Internet connection to the outside world.