There really is no generalized party at state levels -- all states are different. The discrepancy in state results really comes down to the Republicans' big advantage in voter engagement. There are two systemic tendencies in turnout: it varies directly with both income and age. Republicans benefit from both effects.
There is a bit of a vicious cycle, however, which Democrats should take seriously. Think of political office as tiered. The Presidency is tier 1. The Senate, Governorships and major Cabinet positions is tier 2. The House, Lieutenant Governor, and minor Cabinet positions is tier 3. State Senate is tier 4, State House is tier 5... all the way down to county dog catcher or district school board member at tier 20-something.
Winning at tier n makes you a subsequent favorite at n, a contender at n-1, and a dark horse at n-2. Hence, greater success in the lower tiers tends to migrate slowly upwards -- the bench is deeper, greater competition tends to promote better candidates with more impressive resumes.
The Republicans have been better than the Democrats at leveraging these effects -- in fact, donors like the Kochs have invested in the lower levels in order to build a better organization top to bottom. Conservatives have understood the power of dominating at the local level forever -- hence their seeding of local libraries with tracts by Ayn Rand and others throughout the 70s and 80s in order to create an army of local activists.