In case anyone is interested, now that I have a handy kWh meter attached to my dedicated circuit, I monitored how much electricity was used by the EVSE when I left it plugged into the outlet but not the car. The kWh used rose from 7.12 to 7.17 over 28 hours.
My Killawatt meter draws about 0.01 kWh over about 10 hours even with nothing pugged into it. I suspect the one you have is not that different, so at least some of that is probably from the meter. And, as @bisco said, it's a fraction of a penny a month.
Vampire draw is an interesting topic. Our 2 chargers both have Wi-Fi connections. That means there were always be potential for some consumption even when not plugged in. But then again, these older models (from just 2 months ago!) don't have indicator lights. The newest does... for Power, Wi-Fi, and Charging status. Realistically, the usage is trivial compared to what the car itself will ultimately consume.
I agree. I was mostly curious to see if this contributed to my 7.12 kWh measurement after my first charge, since that was more than I was expecting.