If you wanted to count the number of rivers in each state in a new field you could simply count the delimiter, the dash (-) and add 1 (because each dash separates one river from another it'll be one less dash always than the number of rivers).
Go to you field calculator, select Python and use this expression;
(!riversfieldname!.count('-'))+1
Your calculating field should be short integer in this example and should return a value of 6 in this example.