There are many ways to answer this question. Basically, your choice is to use either 1) a physically-based model 2) a statistical model 3) a blend of the two (Kalman filter).
Since you ask the question here, I assume you opted for 2. Do you have good reasons for that? If so, you need to fit a statistical model to your data, and use it to make predictions. You have 24 stations so a spatiotemporal model is warranted. How wide an area do they cover? How many different climate regimes are there? That will determine how complicated a spatial model you need.
The temporal component of that could well be ARIMA: it's often a reasonable choice. Where did you get the idea that it only applied to seasonal data?
Whatever you do, the atmospheric concentration of $CO_2$ has to be one of your predictors. If your data show a big trend from 1970 to present, $CO_2$ is the most likely culprit. Other indices of regional climate variability may be relevant too; depends on your region: where are your stations located?