It depends where. For example during the recent drought in California, increased pumping by farmers led to hundreds of wells going dry, leaving thousands of people without drinking water. In California, climate change projects drier and longer droughts (and more intense rainfall event) which -if not managed correctly- will continue lowering down groundwater levels.
If you only want to know general potential effects: Article Beneath the Surface of Global Change: Impacts of Climate Cha...
Free full text from the authors here: Article Beneath the Surface of Global Change: Impacts of Climate Cha...
I agree with Mr. Jose Pablo somewhat. Besides that, One of the effects of climate change on groundwater in addition to low rainfall, is the change in the type of rainfall and precipitation. For example, in many areas where it is snowing there is no more snow and it turns into rain. This reduces the chance of water penetrating underground aquifers and the rain turns into runoff or flooding and is quickly out of reach.
That will largely depend on the component of climate change you are considering... The temperature indices? changes in rainfall patterns? desert encroachment?. All these can impact heavily on groundwater drawdown but as far as climate change is concerned, changes in rainfall patterns have the highest impact. However, the most obvious changes in groundwater drawdown are usually due to anthropogenic influences...i.e. increase in the frequency and volume of abstraction due to changes in population dynamics...
Climate change which also been associated with increase in the extreme events may impact Aquifers storage, i.e., ground water’s storage capacity so called as Aquifer may go down. Use of optimum policies, for example - adopting rainwater harvesting has potential to overcome ground water storage fall; may be a way to rejuvenate Aquifers.
Climate Change nevertheless effects the rate of groudwater drawdown. If the climate is hot and dry, water requirement for consumption and agriculture increases and hence the drawdown. When mercury is low, the requirement for water is low and hence minimum drawdown!!