My BMW 2002 starts out good in morning but when it gets warm or hot my transmission failsafe light comes on what can I do

BMW has been using electronically controlled automatic transmissions in most models since the late '80s. These transmission systems employ a transmission control unit which controls electrical solenoids to manipulate the various control and shift valves and functions in the transmission valve-body. When the transmission control unit detects a problem that it "thinks" could be damaging to the transmission or engine, it goes into a fail-safe or "limp" mode. In this mode, the transmission will function in reverse normally, but in forward it will operate in only one gear (whether you have the shifter in D, or any of the lower gears). This will typically be 3rd or 4th gear, depending on the model. It will start in this gear and stay in this gear with no shifting. You would experience this as sluggish acceleration and higher than normal engine speed at about 40-mph and above.
If you must drive the vehicle when the transmission is in the fail-safe mode, accelerate gently and keep the road speed lower than full highway speed.
Earlier models do not employ fault codes and must be diagnosed manually. Late models do employ fault codes and a diagnostic/scan machine can be used to check and read the faults to assist in diagnostics. Note that this is not your common OBD-II engine management fault code tool. This is what A BMW dealer or a properly equipped independent shop would have for full vehicle system diagnostics and coding operations.