Multi-dimensional Knapsack with Minimum Value constraints for Dimensions
Your intuition is right on.  If I understand correctly, when $V_1=\cdots=V_d=0$, then your problem becomes exactly the ordinary knapsack problem.  So, the knapsack problem is a special case of your problem.  That means that your problem is at least as hard as the knapsack problem, i.e., your problem is NP-hard.
You can prove this formally by exhibiting a reduction that converts a knapsack problem to an instance of your problem by setting $V_1=\cdots=V_d=0$; thus any algorithm that can solve your problem can be used to solve the knapsack problem, too.