If you are using Lazarus trunk, you can force SynEdit to draw its own caret (rather than using the system caret.
Go to Tools > Options > Editor > Colors, find the color entry for Caret, and set it to black (or whatever you want, but black would be what the OS has).
a) This will keep blinking
b) But if you use a srceen reader, then the reader will not know where the caret is.