If the trail tread is frozen or dry. => trails open
If the trail tread is thawed and/or muddy => trails closed

It would probably require someone with a sense of how the trails reacted to similar conditions in the past if you don't want to physically check. It shouldn't be too hard to determine if the conditions are suitable at the trail by simply looking at how wet/frozen/thawed/muddy the ground is in a similar geographic location, especially if there has been similar weather.

The conditions we are experiencing right now, a heatwave and major snow melt, are probably the best example of when to close the trails in winter and early spring. Apparently our trail status indicators need to be updated.