Adding quotations at beginning and end of a line using sed

[^$] means "any character except the dollar sign". So saying sed 's/[^$]/"/g' you are replacing all characters with ", except $ (credits to Ed Morton):

$ echo 'he^llo$you' | sed 's/[^$]/"/g'
""""""$"""

To say: match either ^ or $, you need to use the ( | ) expression:

sed 's/\(^\|$\)/"/g' file

or, if you have -r in your sed:

sed -r 's/(^|$)/"/g' file

Test

$ cat a
hello
bye
$ sed -r 's/(^|$)/"/g' a
"hello"
"bye"

sed 's/.*/"&"/' YourFile 

Will do the same using full line as pattern replacement &.

In this case g is not needed because you only have 1 occurrence of the whole line per line (default behaviour of sed reading line by line)