Wait to see if the lawn mower starts up after turning the ignition. For push mowers, wait until after pulling the start cable to see if the mower starts.
Listen closely for any sounds made immediately after trying to start the mower. If the mower does not start, listen for a clicking sound as you try to restart it. A clicking sound is a good indication that you have a bad starter.
Keep track of how many times the lawn mower starts right away or requires a few attempts to start. A dying starter will work some of the time, but not every time.