How can my script determine whether it's being run by bash or dash?

You can make the fact that $BASH_VERSION is blank in dash work for you:

if [ "$BASH_VERSION" = '' ]; then
    echo "This is dash."
else
    echo "This is bash."
fi

You just have to use quotes on the variable BASH_VERSION to use -n

if [ -n "$BASH_VERSION" ];then
 echo "this is bash"; 
else 
 echo "this is dash";
fi

Use /proc/[PID]/cmdline to see what the script is being run with and test for what it contains. The $$ variable will give us the PID of the running shell. Thus we can make a script like this,

#!/bin/bash
if grep -q 'bash' /proc/$$/cmdline ;
then
    echo "This is bash"
else
    echo "This is some other shell"
fi

Here's a test of the same script:

$> bash test_script.sh                                                                                                
This is bash
$> dash test_script.sh                                                                                                
This is some other shell