n/a. The highest average temperature in Galveston is July at 86.2 degrees. The coldest average temperature in Galveston is January at 56.4 degrees. The most monthly precipitation in Galveston occurs in January with 0 inches. The Galveston weather information is based on the average of the previous 3-7 years of data.he Galveston, TX air quality index is a median value which considers the most hazardous air pollutants. The Galveston, TX air quality index is 0.8% less than the Texas average and 4.6% greater than the national average.