What's the base height in pixels that I should be aiming for on my website/web application?
I don't think there's a 'one size fits all' answer unfortunately.
I believe you have to look at your target audience and do a best-guess for the screen resolutions to target.
For example, if you are writing an app to be used on desktop machines in your office, you can take a look at what resolution most people are using, probably somewhere between 1024x768 and 1280x1024 and then account for other windows, or the start menu etc. and go for that.
On the other hand, if you anticipate this app being used mostly on mobile browsers then you will have a much different set of resolutions to target.
If your website is lucky enough to have thousands of users that view it on all sorts of screens and resolutions etc, that's great! because its successful, but with that success means more testing of various resolutions will be required to maintain and grow that success.
Bottom line, target the resolutions that will satisfy the majority of your user base, and test the others accordingly.
For a long time (at least in web years), the question of screen resolution was straightforward: what monitor size is the average user going to have.
That's how sites went from targeting 640x480 to 1024x768. The problem is that at that plateau, this simple formula completely broke down.
Today, between iPads and iPhones, Android phones and netbooks, huge monitors, etc. the sizes are all over the place and not everyone runs their browsers at full screen. So, just because I'm writing this on a 24" monitor that does 2048x1152, doesn't mean the browser window that this site has available is that size, much less even that height.
What I'm seeing a lot is people settling in at something like 950 or so for width for non-phone browsers and betting on getting at least 600 pixels for height. Then a second, mobile version that's less "fixed" in size for phones, etc.