Of course, that isn’t how it played out at all in 1995 and 1996; Bill Clinton was widely viewed as having held the line against the Republican onslaught, although he actually did give substantial ground on taxes and a number of other issues. The budget fight became the focal point of Democrats’ attempt to take back the House and Senate in the 1996 elections.
But the Democrats didn’t actually use the shutdown itself as their main line of attack on Republicans. It was part of it, but the real attacks came over the Republicans’ motivation for the shutdown. Because of the expansive nature of the GOP’s cuts, the Democrats were able to focus on several unpopular portions of the GOP budget: the so-called M2E2 strategy. They commenced a mantra-like repetition of their opposition to Republican attempts to gut “Medicare, Medicaid, Education and the Environment” in favor of a “risky tax scheme” that benefitted the rich.
In other words, in evaluating 1996 as an illustration of what will happen to the GOP today, we probably have to separate the tactic of a shutdown from the substance of what motivates it. And today, the GOP is focused on defunding Obamacare, a law that isn’t particularly popular. For the analogy to 1995-96 to really stick, the GOP will probably have had to try something along the lines of shutting down government to implement the Paul Ryan balance-budget plan.
While public opinion might be against the shutdown tactic, there probably won’t be the same level of outrage against the underlying policy motivation, which is what 1995-96 was mostly about. If Obamacare turns out to be the train wreck some conservatives predict (I have no clue whether it will or won’t), the tactic itself might be viewed as less of a negative.