It's easy to be paranoid with the media coverage that West Nile gets, but it tends to be sensationalized. :) Once you know the facts about WNV, you will see that there isn't anything to worry about.
I am going to give you some information, to put your mind at ease.
~80% of people that become infected with WNV are asymptomatic. The majority of the remaining ~20% will get flu-like symptoms and then clear the virus. 1 out of 150 infected will get more severe symptoms, including neurological symptoms, but they tend to be immunocompromised. I know we have had deaths in the area, but the third patient that died was a transplant recipient. She definitely had a compromised immune system. This is why the severe forms are seen more often in elderly patients. If your children are over the age of 2 and healthy, then they would be most likely asymptomatic, even if they contracted the virus. They could have already had it, even.
Just FYI. Don't let WNV keep you from enjoying the outside. If you would like more information, please let me know. I hope this helps.