Asherton is a city in Dimmit County, Texas, United States. The population was 1,084 at the 2010 census, down from 1,342 at the 2000 census. …Asherton is a city in Dimmit County, Texas, United States. The population was 1,084 at the 2010 census, down from 1,342 at the 2000 census. The estimated population in 2018 was 1,064. U.S. Highway 83 passes through Asherton.