Water, like everything else in life, is about balance. Let's start with the two separate parts of watering you should work on trying to balance.
Your seeds need moisture to germinate. Traditionally, you'd be planting those seeds under soil holding that water for them. With microgreens, we are planting the seeds on top of the soil, so we need to be more vigilant about keeping the moisture contained around them.
This is where your container cover comes in. A good cover will hold that moisture in and help to maintain a steady temperature. I use clear covers that come with the seed starting kits I buy. Seeds like basil need light to germinate, and a clear cover can be invaluable.
Another option is to take a container of the same size and flip it upside-down so it sits on top of your seed container. I've also tried laying newspaper over the container and using plastic wrap. I disliked using newspaper; I had to constantly mist because it seemed to hold in the least amount of moisture.
Once most of your seeds germinate, remove your cover. I like to place it askew on the container to allow air in for a day, then completely remove it.
Choosing a good seed starting mix will help you immensely here. It will hold just enough to replenish your plants without getting too soggy. It will also fight against soil gnats and mold if you over water because it is sterile.
Bottom-up Watering Method. Microgreens are fragile, and will sometimes fall over if you drench them from above with your watering can. The bottom-up watering method can help control getting the microgreens the water they need as needed. To bottom-up water, you simply put your microgreens growing container (with holes) inside another basin container without holes. You then put water in the basin and allow the soil in the growing container to soak up this water. Once your soil is sufficiently moist, pour the rest of the water out of the tray. The container shouldn't sit in water for an extended period of time.
Watering too little has an easy indicator. Your microgreens will start falling over, most likely around the edges of your container first. Watering too much can be a little harder to detect, but I find that my nose can help. Soil that stays soggy can start to sour, and the seeds start to rot.
I fill my watering cans back up after I water to let the chlorine evaporate before I water again. The amount of chlorine in my water may not harm my microgreens, but I have done it as standard practice for my house plants and bonsai for many years now.
Mkono Plastic Watering Can - This can puts out more water than the Bosmere, and is generally my go to for watering. Large opening at the top means it's easy to fill and easy to mix in organic additives to my water if I need them.
GroundWork Misting Sprayer Bottle - Not much to say here except I needed a fine mist and paying above $5 was unacceptable. Lots of choices on Amazon, but most are simply too expensive for what you're getting.
Bosmere V127DB Haws Indoor Plastic Watering Can - There are a lot of small watering cans on the market, but the key here is the rose spout. The water is restricted just enough so that my seeds aren’t disturbed. This used to be my primary watering can until I found the Mkono. The water is really restricted in mine, so I would only use this if your sticking to 5x5 or smaller containers.
Nothing - If your using the bottom-up watering method, you could use almost anything to water since you would only need to pour water into the larger container holding the water.