Is K-NN applicable for binary variables? I need help because I'm just new to machine learning and I do not know if k-nearest neighbors algorithm can be used to identify the appropriate program(s) for Student 11 in the table below. 

The school subjects (Math, English, etc.) are the features, while the 'Program' column has the class labels A, B, and C.
The binary values represent the interest of a student in the subjects.
Can K-NN algorithm find the similar students based on the binary variables in the table?
Yes, you just have to find a suitable distance metric, instead of using the default Euclidean distance. Euclidean distance will work, but it loses a lot of its positive points when used on a non-euclidean space.

For you specific case, the Jaccard distance basically measures how many 1's are equal on both instances, ignoring the dimensions where both are 0's. This gives an interpretation like "if a program has a course, but the other doesn't, then they are dissimilar". Jaccard index is very useful in high-dimensional boolean matrices, such as generated by one-hot encodings.

Other more intuitive choice is the perfect match distance, which would simply measure how many dimensions are different for the two instances and can be easily computed by $\sum_{i=1}^n |X_i - Y_i|$. In this case, the interpretation becomes similar to "if both programs have a course, or both program don't have a course, then they're similar".

However, be careful with the K value of your K-NN. You only have two instances of class B, so you will have to choose 1 or 2 (technically at most 3) for the value of K.