Several food-borne pathogens are spreading due to climate change. Depending on the crop type, drought and excess moisture encourage crop pests and spread molds. Climate-driven rusts, blights, and rots are devastating already stressed crops and thereby indirectly affecting human health through decreased food security.
But it's not just our crops falling ill from increased pathogens in a changing climate. Instances of some types of pathogens that make their way to people are also on the rise. The US Global Change Research Program's Climate and Health Assessment from 2016 describes some of these pathways and consequences (and in their bulleted list [2], they include ample citations to support each bullet point - I'm linking out to that directly in hopes the links there are updated as necessary):
Water-borne diseases are closely related to weather and climate. Cholera, cryptosporidiosis, and several other significant diseases are spread by fecal contamination of water supplies and are often closely associated with floods and heavy downpours. In other cases, drought can concentrate disease pathogens in pools and low flows. Climate change is causing increased intense rainfall events in many parts of the world and drought in others; it is not surprising that water-borne illness is a growing problem in those areas with the most extreme changes. The key to understanding the spread of water-borne illness in a changing climate is exposure. As this image shows, various factors (all from human activity) are increasing our exposure to these illnesses.
Links
[1] http://www.fao.org/tempref/docrep/fao/011/i0378e/i0378e.pdf
[2] https://health2016.globalchange.gov/food-safety-nutrition-and-distribution#table-125
[3] http://climateandhealthalliance.org/uncategorized/climate-change-and-food-safety/
[4] https://health2016.globalchange.gov/water-related-illness