Well, the instrument of strike is not really blunt. A pilot strike can put some real pressure on an employer, and an angry public certainly helps to increase this pressure.
Some uneducated people that don't (want to) understand what pilots are fighting for are what I would consider collateral damage.
Of course, the press tries to whip it up against the oh-so-overpaid-get-tons-of-money-for-lazily-pushing-some-buttons pilots and tries to stir anti-pilot sentiments, but a lot of sensible and informed folks do get the pilots' points.
Compared to strikes e. g. in the IT sector (which I haven't heard about yet, but not all is going well there either with all that outsourcing to Asia and cost cutting), pilots are in a much better position to lend weight to their issues.
If IT guys went on strike, well ... hardly any employer would seriously care.