Use

    StringReplace[list, PunctuationCharacter -> " "]

    (* {"string1", "bla bla", "more stuff", "and more stuff"} *)

[`PunctuationCharacter`](http://reference.wolfram.com/mathematica/ref/PunctuationCharacter.html) 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"} *)