You could accomplish this by arranging for HISTFILE to be unset when the user logs in:
From the Bash man page:
The name of the file in which command history is saved (see HISTORY below). The
default value is ~/.bash_history. If unset, the command history is not saved when an
interactive shell exits.
Alternately, you could set HISTFILE=/dev/null in ~/.ssh/environment, after ensuring PermitUserEnvironment is set in /etc/ssh/sshd_config.
Yet another option is to add
in your ssh account's ~/.bashrc.