Political parties make attempts to get/keep themselves in power. Addressing issues is a side-effect of those attempts, or the work of idealistic individuals who only joined a party because that's how you get enough money to get elected. However, I'll play this game with you.
I don't see either party doing anything meaningful about mental illness. You're asking me to choose between the greater of two non-existent entities. As I see it, the general every-man Repub response is "identify them and keep them off the streets until they can be treated beyond a shadow of a doubt, if they can be treated beyond a shadow of a doubt". The general every-man Liberal response seems to be "find a label to give them, make concessions for having that label, treat them until you decide they can't be treated, then give them a different label and treat for that, keep them on the streets until they hurt someone." Neither is particularly palatable. For the safety of society, I'd probably go the Repub route. For the good of the individual we're talking about, the Dem way seems much better. Neither way seems particularly effective at addressing both. Repubs spout on about not allowing government to infringe on individual liberties, but fail miserably in this case, while Dems cry "get rid of guns for public safety, while we allow identifiable psychopaths to walk among you," which is another miserable fail.
Economic standing is supposed to be a transient state. It should be true that if you are poor you would work hard, get good at your career, make more money and end up better off financially than when you started (middle class, maybe even rich). Social programs designed to make it easier on the poor appear to have entrenched them as poor. Poverty appears to be less transient than it once was. My father (old enough to associate a stigma with social welfare programs) and his father before him (before social welfare programs) had to bust their ***** working 60+ hour weeks just to provide necessities and a few niceties. Until later in their careers, when they were able to leverage their experience and skills for more money. Now the incentive isn't there unless you're supremely self-motivated. The necessities and a few niceites are provided for you, regardless. If you're asking which party offers a better path out of poverty, I'll say the Republicans. If you're asking which party makes poverty more liveable when you're there, I'll say the Democrats.