P. Halmos in "I want to be a mathematicien" has a short section on refereeing where he exposes his point view: The role of the referee in his eyes is not to certify correctness of a paper (this is the authors job, according to Halmos) but he has to "smell" a paper and to advise the editors on its interest (I am citing from memory and hope that there is not too much distorsion).
As a referee, I am following his advice in the following sense: If I enjoy reading a paper then I recommend it generally for publication (and in this case I check also more or less carefully the proofs). If I can find no pleasure and no interest then I suggest either another referee if the paper seems interesting nevertheless or I recommend rejection. In the last case, I do generally not check proofs (and I tell the editor and the authors so).
In some sense, mathematics should mainly be interesting, errors in very interesting and stimulating papers can be tolerated to some extend since they will generally get quickly corrected. (This is of course only true for exceptional papers, most papers have close to zero readers anyway.)
Mathoverflow is somehow a mirror of the remaining mathematical litterature (except that the process is much faster on MO): Erroneous statements get quickly commented and eventually corrected or retracted.
I believe that the possibility of making errors is a necessary part of any creative process. Computers make (generally) no errors. They are also more stupid than the most disabled human who is not braindead.