You want
sed --in-place 's/[[:space:]]\+$//' file
That will delete all POSIX standard defined whitespace characters, including vertical tab and form feed. Also, it will only do a replacement if the trailing whitespace actually exists, unlike the other answers that use the zero or more matcher (*).
--in-place is simply the long form of -i. I prefer to use the long form in scripts because it tends to be more illustrative of what the flag actually does.
It can be easily integrated with find like so:
find . -type f -name '*.txt' -exec sed --in-place 's/[[:space:]]\+$//' {} \+
If you’re on a Mac
As pointed out in the comments, the above doesn’t work if you don’t have gnu tools installed. If that’s the case, you can use the following:
find . -iname '*.txt' -type f -exec sed -i '' 's/[[:space:]]\{1,\}$//' {} \+