Ours is one universe among an enormous number
The objection
If there are vastly many universes with different constants, then some of them will have life-permitting values, and of course we find ourselves in one of those. The idea is not invented to escape design; it falls out of physics people were doing anyway. Eternal inflation produces bubble universes as a consequence of inflationary cosmology, and string theory's landscape allows an enormous number of possible low-energy configurations. On this picture, asking why our constants are life-permitting is like asking why the planet you live on has liquid water. Somebody had to be on one that did.
The response
This is the main response to fine-tuning and it is a serious one, held by serious physicists for reasons internal to their field. Its honest limit — and physicists say this themselves — is that there is currently no empirical evidence for other universes and no accepted way to test for them. A hypothesis that explains the data but cannot presently be checked is not thereby false; it is a promissory note. Two further points, and one of them cuts against this page's own side. Against the multiverse: some philosophers press the inverse gambler's fallacy — that many other trials elsewhere does not raise the probability of this particular trial coming out well, any more than learning that many people play the lottery explains your own winning ticket. In favor of fairness: a multiverse would in fact explain the tuning, and a Christian who waves it away as desperate is not arguing honestly. Where this lands: two explanations of the same data, one of which posits an unobserved designer and one of which posits unobserved universes. Neither side gets to call the other unscientific for free.