Nuclear track detector CR-39 is made of is a sample of a solid material, so if we irradiate this detector by He-Ne laser so what is the effect of that on track detector properties?
Absolutely no any effect. There is no absorbtion at 632 nm. And the power of He-Ne laser is too low to expect any nonlinear effects. So you can try to use He-Ne laser for track visualisation.
Laser can affect the physico-chemical properties of CR-39, which one can measure by spectrophotometric techniques such as UV-Vis (where you can find band-gap from Tauc's plot), FTIR (to see the behaviour of functional groups).