it is not possible to use neutrinos in a transistor as the caller described. they are nearly massless, have no electric charge, and only affected by gravity and the weak force (the force behind radioactive decay). neutrinos interacting with other matter is a rare occurrence, and the description is linked below.
normal transistors work by changing the electric charge of the gate or base (a part of the transistor) usually be applying (or by stopping the application of) electricity.
edit:
please note that there are as many as About 100 trillion neutrinos pass through our bodies every second.
i purposely did not explain the weak force, the electro-weak theory, and a large number of other physics related topics. i didnt want to “wander off into the weeds” as it were.
all of those can be googled as you wish.