Why PNP transistor might be opened even though (almost) nothing is connected to the base?
Could be leaky, could be pickup of AC hum from the mains or RF from AM radio etc. as the transistor will act as a detector.

To tell the difference, put a 1uF ceramic (low leakage) capacitor from base to emitter and see if the issue disappears. 

In any case, you can reduce the sensitivity simply by connecting a relatively high-value resistor from emitter to base. Something in the range 10K-100K will likely work for you. 

Edit: One thing that can cause a great deal of leakage is the use of acid-core (plumber's) solder. If this is accidentally used, clean the board with warm soap and water, preferably followed by isopropanol alcohol. I have seen mains circuits burn up as a result of this. As an aside, the solder used in plumbing was often 50:50 PbSn, so it would not solder as nicely (and will require much higher temperatures) compared to good electronic grade 63:37 solder.