It's inherently impossible to do exactly this, because a command (or pipeline of commands) produces output as it runs, but doesn't produce an exit status (success/failure) until it's finished running; therefore, you can't decide whether to save the output or not until it's already finished being output.

What you can do is store the output somewhere temporary, and then either save that or not. I'm not sure if this is quite what you're trying to do, but maybe something like this (using a variable as the temporary storage):

while true
do 
    output=$(ss --tcp --processes | grep 53501)
    if [ -n "$output" ]; then
        echo "$output" >/tmp/cmd.out
    fi
done

The while loop looks dangerous, eventually you will run out of disk space.

while true
do 
ss --tcp --processes | grep 53501 &>> /tmp/cmd.out
sleep 1
echo "Careful about using while true without any sleep"
done

&>> pipes and appends all STDERR and STDOUT to file, naturally the output will be nothing if grep finds nothing.