I know this is old, and all these solutions are hacks around browser security precautions with real value.
That said, as of today, fileInput.click() works in current Chrome (36.0.1985.125 m) and current Firefox ESR (24.7.0), but not in current IE (11.0.9600.17207). Overlaying a file field with opacity 0 on top of a button works, but I wanted a link element as the visible trigger, and hover underlining doesn't quite work in any browser. It flashes on then disappears, probably the browser thinking through whether hover styling actually applies or not.
But I did find a solution that works in all those browsers. I won't claim to have tested every version of every browser, or that I know it'll continue to work forever, but it appears to meet my needs now.
It's simple: Position the file input field offscreen (position: absolute; top: -5000px), put a label element around it, and trigger the click on the label, instead of the file field itself.
Note that the link does need to be scripted to call the click method of the label, it doesn't do that automatically, like when you click on text inside a label element. Apparently the link element captures the click, and it doesn't make it through to the label.
Note also that this doesn't provide a way to show the currently selected file, since the field is offscreen. I wanted to submit immediately when a file was selected, so that's not a problem for me, but you'll need a somewhat different approach if your situation is different.