Batman has always been apolitical to me. However, were I to examine the issue, I would suggest Batman is very much not a Republican, nor Democrat. His very nature as a grass-roots vigilante really removes him from either party by ideology. However, let's examine the views Batman has expressed over the years (in-continuity) against political affiliations:
Gun Rights: Batman has been depicted for numerous years as being a staunch anti-gun individual. In short, Batman hates guns and thinks they should be eliminated from the equation of our culture. Definitely a Democrat viewpoint.
Government interference: Batman has been shown infiltrating and expressing disdain for government operations, mostly in connection with law enforcement (FBI, etc.). Also, Batman's whole ideology represents a citizen's right to defend himself and pro-actively thwart crime. Very much a classical Republican viewpoint (recent Republican stances seem to go against this view)
Economy: Bruce Wayne has shown a remarkable philantropic attitude, giving to charities and welfare organizations over the years, as well as calling for government help in the rebuilding of Gotham after Cataclysm. Definitely a Democratic viewpoint.
Defense: Batman is definitely a supporter of the armed forces and enforcement of the protection of America. I would assume he would support a large spending of the budget on military protection. Republican.
There's more, but I have work to do