$ sed -i '/^$/d' <name_of_your_textfile.txt>
Or, if the "blank lines" are allowed to contain tabs and spaces:
$ sed -i '/^[ \t]*$/d' <name_of_your_textfile.txt>
-i stands for "edit in place", so if your text files are valuable you might want to use the commands without the
-i and pipe the output to a different file, look at the results and then rename it.
Using Vim (should come with Cygwin, but there is also a native Windows version):
This has the advantage that you directly see your results and can undo them by pressing "u".
You also might encounter fewer problems with Windows' line endings (depends on your Cygwin configuration).