Use
StringReplace[list, PunctuationCharacter -> " "]
(* {"string1", "bla bla", "more stuff", "and more stuff"} *)
PunctuationCharacter is new in version 10.3. The 10.x updates received many new functions for text processing. In older versions I would use an explicit list of possible punctuation characters, i.e.
punctuation = Alternatives@@Characters[",.?:;'\"!-"]
I'm pretty sure I missed some.
StringReplace[list, punctuation -> " "]
(* {"string1", "bla bla", "more stuff", "and more stuff"} *)