Test if a string contains only whitespace and non-alphanumeric characters

Both your regex and syntax is incorrect. You need not use regex at all, just need glob for this:

checkStr() {
   [[ $1 != *[a-zA-Z0-9]* ]]
}

This will return false if even one alphanumeric is found in the input. Otherwise true is returned.

Test it:

$> if checkStr 'abc'; then
   echo "true case"
else
   echo "false case"
fi    
false case

$> if checkStr '=:() '; then
   echo "true case"
else
   echo "false case"
fi
true case

$> if checkStr '=:(x) '; then
   echo "true case"
else
   echo "false case"
fi
false case