Dengue fever has been a threat to public health not only in tropical regions but non-tropical regions due to recent climate change. Motivated by a recent dengue outbreak in Japan, we develop a two-patch model for dengue transmission associated with temperature-dependent parameters. The two patches represent a park area where mosquitoes prevail and a residential area where people live. Based on climate change scenarios, we investigate the dengue transmission dynamics between the patches. We employ an optimal control method to implement proper control measures in the two-patch model. We find that blockage between two patches for a short-term period is effective in a certain degree for the disease control, but to obtain a significant control effect of the disease, a long-term blockage should be implemented. Moreover, the control strategies such as vector control and transmission control are very effective, if they are implemented right before the summer outbreak. We also investigate the cost-effectiveness of control strategies such as vaccination, vector control and virus transmission control. We find that vector control and virus transmission control are more cost-effective than vaccination in case of Korea.