By default, the command prompt uses keys similar to emacs(1); however, if the VISUAL or EDITOR environment variables are set to something containing vi (such as vi or vim or nvi), then vi(1)-style keys are used instead.
The implication is that adding EDITOR=vim to your Bash/.bash_profile or related file on a remote server is the easiest way to kick tmux into vi mode permanently. Or, if you can't edit that, export EDITOR=vim.