How can I tell if I'm in a tmux session from a bash script?

Tmux sets the TMUX environment variable in tmux sessions, and sets TERM to screen. This isn't a 100% reliable indicator (for example, you can't easily tell if you're running screen inside tmux or tmux inside screen), but it should be good enough in practice.

if ! { [ "$TERM" = "screen" ] && [ -n "$TMUX" ]; } then
  PS1="@$HOSTNAME $PS1"
fi

If you need to integrate that in a complex prompt set via PROMPT_COMMAND (which is a bash setting, by the way, so shouldn't be exported):

if [ "$TERM" = "screen" ] && [ -n "$TMUX" ]; then
  PS1_HOSTNAME=
else
  PS1_HOSTNAME="@$HOSTNAME"
fi
PROMPT_COMMAND='PS1="$PS1_HOSTNAME…"'

If you ever need to test whether tmux is installed:

if type tmux >/dev/null 2>/dev/null; then
  # you can start tmux if you want
fi

By the way, this should all go into ~/.bashrc, not ~/.bash_profile (see Difference between .bashrc and .bash_profile). ~/.bashrc is run in every bash instance and contains shell customizations such as prompts and aliases. ~/.bash_profile is run when you log in (if your login shell is bash). Oddly, bash doesn't read ~/.bashrc in login shells, so your ~/.bash_profile should contain

case $- in *i*) . ~/.bashrc;; esac

As for previous answers, testing the ${TERM} variable could lead to corner cases, tmux sets environment variables within its own life:

$ env|grep -i tmux
TMUX=/tmp/tmux-1000/default,4199,5
TMUX_PANE=%9
TMUX_PLUGIN_MANAGER_PATH=/home/imil/.tmux/plugins/

In order to check if you're inside a tmux environment, simply check:

$ [ -z "${TMUX}" ] && echo "not in tmux"

After trying different ways, this is what ultimately worked for me, in case it helps anyone out there:

if [[ "$TERM" =~ "screen".* ]]; then
  echo "We are in TMUX!"
else
  echo "We are not in TMUX :/  Let's get in!"
  # Launches tmux in a session called 'base'.
  tmux attach -t base || tmux new -s base
fi

In this code snippet, I check to see if we're not in TMUX environment, I launch it. If you put this code snippet in your .bashrc file, you will automatically run TMUX anytime you open terminal! P.S.: tested on Ubuntu shell.