Since I'm now getting angry emails about things I've never even said let's clear a few things up.
I'm not hostile to religion. I don't much care about religion. I'm not much interested in it. This isn't strange. Most people aren't much interested in religion other than their own, if that.
I'm not sick of religious people. I think it's great that they're free to believe and practice their religion in any way they want. I'd like to keep it that way.
I am sick of people who keep claiming that the Democratic party is hostile to religious people and controlled by secular liberals who are hostile to religion. If by "Democratic party" you mean "some people who post anonymous comments on the internet" you may have a point. Otherwise, the idea is ludicrous.
Do the Democrats have a perception problem about religion? Sure. We have a political party which has been claiming to be God's Own Party for decades. We have a mainstream media which equates Christian with Religious Right most of the time, and news anchors who don't think liberals can be "good Catholics." We also have some left-leaning Christians who seem to think this perception problem is due to hostility to religion by secular liberals who (see below) have no public presence. I don't understand this. People who perpetuate right wing talking points about Democrats always piss me off especially when they have no basis.
Secularism has essentially no representation in our media or politics. I'm sure there are secular politicians and media types, but few discuss it. No one gets on tv or writes newspaper columns or in any way participates in our contemporary mainstream political discourse and praises secularism or atheism or anything similar, and certainly not in a way which denigrates religious beliefs generally. Advocates for the separation of church and state are not advocating secularism, aside from government secularism, they're simply trying to defend freedom of religion.
Can Democrats appeal to evangelical voters by doing X? Sure, and they can appeal to [insert voting bloc here] by doing Y. The question is can they do so without alienating lots of other people or compromising their principles. Maybe they can. I have no idea. But that's politics. I'm happy to hear about ways to reach out to religious voters, though not being a politician it isn't actually my job to do so. If people vote Republican because they perceive some guy on the internet with no actual official relation to the Democratic party in any way is insufficiently deferential to their religious beliefs then I'm really not sure what I can do about that. I don't really require people to be deferential to my beliefs.
Moderate/swing/independent voters respond to personal charisma and the perception that somebody "knows what they stand for." You know, spine, backbone, etc. I'm sure some genuinely religious politicians can use their faith to help send this message. There are lots of other politicians who can find other ways to do so.
From a policy perspective I'm personally not really interested in compromising on sex or reproductive rights in order to get votes. As with every other issue of course messaging can be improved, though I'd rather focus on getting the "pro-choice for me but not for thee" crowd to understand that they are, in fact, pro-choice whether they know it or not rather than talking about how icky abortion is. I don't know why the public face of religion in this country is concerned with almost nothing but sex, but I'm not really sure those people can be reached.
As with Democrats who constantly fret that they aren't seen as "tough" enough, people who fret that Democrats are not "religious" enough simply reinforce the perception without improving it. Want to be seen as tough? Stop being a WATB and start being tough. Want to make the religious left a visible force in this country? Go on TV and or write columns or whatever which, instead of whining about how awful everyone else on the left is on religion, reassert the public face of the religious left in this country.
