At least in California, everyone I know was taught about the dust bowl during early elementary school years. I honestly don’t remember most what history was taught back then but it seems like such a random event to teach to us when we’re that young. I get that it’s probably cool for the kid, but it’s never been brought back up for me in more detail later in school.
A possible explanation for this would be how the Dustbowl effected California even though it did not physically touch California, if you get what I mean. A popular depiction of these events would be Steinbeck's Grapes of Wrath. This work of fiction in many ways establishes the bones of the how and why California was affected, but is by no means a great source.
One of the main effects of the Dustbowl on Californians was the massed migration of farmers and laborers from the Plains states to the West. These were mostly tenant or sharecropper farmers who had to or chose to give up their leases and land after crop failures. The general accepted number of migrants that moved west is between 700,000 to about 1,000,000, of which it is believed about 300,000 settled in California. They would take menial jobs as pickers and packers in fruit orchards. This would have arguably been one of the largest migration events Californians had experienced since the Gold Rush in 1849. This would have been important in changing the political and cultural landscape in California due to the large influx of migrants.
California was also less impacted by the ecological and environmental events that caused the Dustbowl, in particular, California was not as heavily hit by drought as the Midwest/plains states and also was better able to utilize irrigation. Additionally, and why there is still a large amount of agricultural production in California, the growing season is much longer in California leading to more stable work for these migrants.
This massed migration would later be termed the “Okie” migration base on a generalization that the majority of migrants came from Oklahoma. Steinbeck’s novel also helped to reinforce this idea in the years after the Depression. Everything would not be smooth sailing for the migrants either, as many of the Western states did not want to deal with the sudden influx of workers. In many areas “bum blockades” were established in order to stop these workers from entering states or counties. This was generally done in an attempt to stop wage prices from plummeting. Even with the cultural influx that occurred there was still plenty of prejudice and discrimination against the “Okies” in states like Colorado and California. It was not all smooth sailing. So the main answer would be how influential this event was to Californians as many families are probably descended/related to the migrants. It was a culturally important event. Hope this helps in some way.
Sources: n.d. Gary C. Anderson, Kathleen P. Chamberlain. 2007. Power and Promise: The Changing American West / Edition 1. Pearson.