Finding all the subsets in an array of integers that sum to a given target
Violations of general good practices


Using String.substring to extract single characters is odd. Use String.charAt instead next time
Parsing single character strings using Integer.parseInt is odd. Use String.charAt(index) - '0' instead
allSubsets should be declared as Set instead of HashSet (use interface types instead of implementations)
Method names should be camelCase
No need to write JavaDoc for private methods, simple comments are good enough, if needed at all


Strange way to collect integers

The way you collect integers in a String is strange in many ways:


The program will not work for numbers > 9
Appending digits to a string one by one, and then finally parsing them from the string one by one is strange. 


You could use a Set instead. Sure, maybe it seems a bit strange and inefficient to clone the set in each recursive call, but I still think it would be better. It would also solve the problem you found, with the duplicate sequences in the results.
Probably there is an even better way, but I have no more time now.

Strange static and non-static variables

allSubsets is static, but input and target are not.
It would make more sense to either make all of these static, or none of them.

Not suggested implementation

I can't say I suggest this implementation, but it solves some issues with yours:

private void find(int[] input, int target, Set&lt;Integer&gt; ramp, int index) {
    if (index &gt; input.length - 1) {
        if (getSum(ramp) == target) {
            allSubsets.add(toString(ramp));
        }
        return;
    }

    find(input, target, updatedSet(ramp, input[index]), index + 1);
    find(input, target, ramp, index + 1);
}

private Set&lt;Integer&gt; updatedSet(Set&lt;Integer&gt; ramp, int integer) {
    Set&lt;Integer&gt; updated = new HashSet&lt;Integer&gt;();
    updated.addAll(ramp);
    updated.add(integer);
    return updated;
}

private String toString(Set&lt;Integer&gt; ramp) {
    StringBuilder builder = new StringBuilder();
    for (Integer integer : ramp) {
        builder.append(integer);
    }
    return builder.toString();
}

private static int getSum(Set&lt;Integer&gt; integers) {
    int sum = 0;
    for (Integer integer : integers) {
        sum += integer;
    }
    return sum;
}