1 pointOr, if you are just looking for number of RECORDS (as you say in the title of your posting) -- use Text File Begin Process, which transfers the records one at a time into a variable, and you can add to a counter each time through.
1 pointThe easiest way I found was: Create the special character Carriage Return or Line feed. (Newline) Copy the text of the first variable to a second variable. Remove all instance of the CR or LF from the second string variable. Subtract the number of characters of the second string variable from the original variable. The difference is the number of newlines. Add 1 to that value. You may want to trim the original text first to ensure there is not a trailing newline set.