What kind of decoupling capacitors are you using?
One of the effects of bringing a multimeter probe (or your finger, or anything like that) close to the power pins of a PIC is adding additional capacitance. If you don't have enough capacitance on there already, this can sometimes make the difference. at the very least, you should have a small ceramic cap like 0.1uF directly across the power pins as close to the PIC as possible, and even better, have one on each side of the PIC, and a small electrolytic across the power rails elsewhere.
Probably 3/4 of the times that I have experienced inconsistent/unstable programming like this, it has been due to forgetting sufficient decoupling caps, most importantly the small ones directly next to the PIC.