How to tell if a repaired laptop power cord adapter is safe?
The advice you already got to use a voltmeter is good. 

Check that the DC voltage is correct and also make sure there isn't a significant AC component to the voltage (for reference, my third party laptop power adapter has just under 3 volts AC unloaded). You are correct that the repaired adapter could be dangerous. It's likely only dangerous in another fault condition though. I've seen people short blown fuses with a wire as opposed to replacing them. This could cause damage to the laptop or, in the worst case, start a fire were there to be another fluctuation. 

Personally, if the voltage matches up, I wouldn't be too uncomfortable using the repaired adapter. Still, it might be good to get a hold of the person and ask them how they fixed it.