April Aire 600 powers off while furnace is firing
Humidifier must run while the furnace runs

Aprilaire model 600 operates on the principle of evaporation. It's an evaporative cooler ("swamp cooler") installed in the heating system. It's installed as a bypass between the furnace outlet and the cold air return; a portion of the hot air coming out of the furnaces goes through the Aprilaire 600, picks up moisture, and goes immediately into the furnace air return.

Because of its operation, it's very important that the water valve can operate while the furnace blower and burners are running. Otherwise, each heating cycle of the furnace can add no more water than whatever is left standing in the humidifier's pad.

ecobee could control the humidifier

The ecobee3 and ecobee4 thermostats can control that Aprilaire 600 directly. Is yours one of these? Do you prefer to use the Aprilaire controller instead, or is this arrangement simply a result of not knowing the Ecobee has this capability? It's not impossible to make everything work correctly with the Aprilaire controller but it's simpler to let the ecobee manage humidity for you.

Wiring for ecobee control of humidifier

When the ecobee is installed there's a setup wizard that figures out whether and how you have a humidifier connected. After getting the wiring sorted out we'll circle back to fixing that configuration (I'll edit it in after you confirm this is the direction you want to go).

If your ecobee model is one that supports accessories there's an easy way and a less-easy way to wire it. The easy way involves the transformer that came with the Aprilaire unit. Make a ring: one transformer wire connects to one water valve wire, the other water valve wire connects to one of the ecobee ACC terminals, the other ecobee ACC terminal connects back to the second transformer wire.

The less-easy way powers the Aprilaire 600 water valve using the transformer that's already in the furnace. This is how I have mine set up, but I don't recall exactly which wires I used. I'll have to edit to add detail on that..