Finding the number of vowels in a string
Outline

In this case, instantiating a new CountVowels() is overkill.  all you need is a function.

As a rule, classes should be named as nouns.  A better name for the class would be VowelCounter.

public class VowelCounter {
    public static int vowelCount(String s) {
        …
    }

    public static void main(String[] args) {
        …
        System.out.println(vowelCount(userInput));
    }
}


Implementation

Your countingVowels() function is fine, but could be simpler.

public static int vowelCount(String s) {
    int count = 0;
    for (char c : s.toLowerCase().toCharArray()) {
        if ("aeiou".indexOf(c) &gt;= 0) {
            count++;
        }
    }
    return count;
}


Specifically, note:


Using an enhanced for loop is less tedious than a counting loop.
Avoid boxed types such as Character, when a primitive char will do.
For such a short array, a binary search is overkill.




Your main() is also OK, but I would write it this way:

public static void main(String[] args) {
    try (Scanner scanner = new Scanner(System.in)) {
        String userInput = scanner.next();
        System.out.println(vowelCount(userInput));
    }
}


In particular:


A Scanner is AutoCloseable, so a try-with-resources block is tidier than calling close() on it manually.
There doesn't seem to be much point in the …1 suffix on your variable names sc1, cv1, and s1.