What brand is your water heater and model ?
When the gas switch is turned on , my Suburban 10 gal heater the red indicator light only stays on until the flame is confirmed , then it stays off unless flame fails to ignite when the water temp drops .
Have you checked the position of the water heater by-pass valves ( winterizing valves ) do you have an outdoor shower , are the shower valves there fully off .
Do you have the users manual for your water heater ?
If not free down loads at this site .
www.myrvworks.com/manuals/