Over the past years, the application of deep neural networks in a wide range of areas is noticeably increasing. While many state-of-the-art deep neural networks are providing the performance comparable or in some cases even superior to humans, major challenges such as parameter settings for learning deep neural networks and construction of deep learning architectures still exist. The implications of those challenges have a significant impact on how a deep neural network is going to perform on a specific task. With the proposed method, presented in this paper, we are addressing the problem of parameter setting for a deep neural network utilizing swarm intelligence algorithms. In our experiments, we applied the proposed method variants to the classification task for distinguishing between phishing and legitimate websites. The performance of the proposed method is evaluated and compared against four different phishing datasets, two of which we prepared on our own. The results, obtained from the conducted empirical experiments, have proven the proposed approach to be very promising. By utilizing the proposed swarm intelligence based methods, we were able to statistically significantly improve the predictive performance when compared to the manually tuned deep neural network. In general, the improvement of classification accuracy ranges from 2.5% to 3.8%, while the improvement of F1-score reached even 24% on one of the datasets.