There are two ways to stop the blinking cursor on the IIe.
1. If you have an 80-column card, do a PR#3 and then an ESC 4. This uses the 80-column firmware but in 40-column mode.
2. If there is no 80-column card installed on your system, do a POKE 49162,0:PRINT CHR$(4)"PR#3". This has the same effect.
In the 80-column firmware, the cursor is an inverse blank. This effectively removes the flashing cursor.