Faster and more cost-efficient, crowdsourced delivery is needed to meet the growing customer demands of many industries, including online shopping, on-demand local delivery, and on-demand transportation. The power of crowdsourced delivery stems from the large number of workers potentially available to provide services and reduce costs. It has been shown in social psychology literature that fairness is key to ensuring high worker participation. However, existing assignment solutions fall short on modeling the dynamic fairness metric. In this work, we introduce a new assignment strategy for crowdsourced delivery tasks. This strategy takes fairness towards workers into consideration, while maximizing the task allocation ratio. Since redundant assignments are not possible in delivery tasks, we first introduce a 2-phase allocation model that increases the reliability of a worker to complete a given task. To realize the effectiveness of our model in practice, we present both offline and online versions of our proposed algorithm called F-Aware. Given a task-to-worker bipartite graph, F-Aware assigns each task to a worker that minimizes unfairness, while allocating tasks to use worker capacities as much as possible. We present an evaluation of our algorithms with respect to running time, task allocation ratio (TAR), as well as unfairness and assignment ratio. Experiments show that F-Aware runs around 10^7 x faster than the TAR-optimal solution and allocates 96.9% of the tasks that can be allocated by it. Moreover, it is shown that, F-Aware is able to provide a much fair distribution of tasks to workers than the best competitor algorithm.