The general rule is to try to match the size of the entrance to the size and activity of the colony. If the bees are bumping into each other at the entrance, it's a sure sign that it should be wider. However, during a honey flow combined with high temps, it should be open all the way. This is not only to make it easier for the bees to cool off the hive (thereby avoiding one of the conditions that lead to swarming) but also to make it easier to evaporate the water in the nectar they are bringing in. [Actually, both of these are related, since evaporation of water cools the hive].
If I started from a package or swarm should I keep the entrance reduced until I get new brood emerging? The way I figure from install for the first ~28 days the population will dwindle through normal death rates and once there is emerging brood then the population will start increasing again. I'm at day +10 from package install and +7 from swarm install.
DLMKA asks: "should I keep the entrance reduced until I get new brood emerging?"
Efmesch expresses his opinion (I'm sure there are others): I don't think so. The entrance should be suited for the active traffic. Unless you have problems with robbing, it need not be kept at as reduced an opening as possible. A bit oversized is much better than a bit undersized. The entrance isn't something you have to adjust frequently. Two to three times a year should be enough (at least in my climate and my beekeeping conditions).