Decentralized dual methods play significant roles in large-scale optimization, which effectively resolve many constrained optimization problems in machine learning and power systems. In this article, we focus on studying a class of totally non-smooth constrained composite optimization problems over multi-agent systems, where the mutual goal of agents in the system is to optimize a sum of two separable non-smooth functions consisting of a strongly-convex function and another convex (not necessarily strongly-convex) function. Agents in the system conduct parallel local computation and communication in the overall process without leaking their private information. In order to resolve the totally non-smooth constrained composite optimization problem in a fully decentralized manner, we devise a synchronous decentralized dual proximal (SynDe-DuPro) gradient algorithm and its asynchronous version (AsynDe-DuPro) based on the randomized block-coordinate method. Both SynDe-DuPro and AsynDe-DuPro algorithms are theoretically proved to achieve the globally optimal solution to the totally non-smooth constrained composite optimization problem relied on the quasi-Fejér monotone theorem. As a main result, AsynDe-DuPro algorithm attains the globally optimal solution without requiring all agents to be activated at each iteration and thus is more robust than most existing synchronous algorithms. The practicability of the proposed algorithms and correctness of the theoretical findings are demonstrated by the experiments on a constrained Decentralized Sparse Logistic Regression (DSLR) problem in machine learning and a Decentralized Energy Resources Coordination (DERC) problem in power systems.