I haven't yet have a device damaged by a leaking alkaline battery. I had some destroyed by leaking cheap zinc-carbon batteries though. Some alkaline batteries did leak, but I just wiped away the powder and/or liquid. In one case I even continued to use the already leaking batteries until empty because they still had juice in them. I just took care of wiping off whatever came out of them regularly. These were an old set of 6 Duracell Ultra "D" cells which I used as an interim solution in the summers of 2009 and 2010 before getting the rechargable LSD's I use now. Those Ultras actually were bought back in 2000 as an emergency for my portable keyboard when I accidentally forgot its power supply at home, but, although they are Ultras, they will only power the keyboard for at most 2-3 hours, using up only 20% of their total energy (while the keyboard can be run on the LSD D cells for up to 8 hours without problems... at least I had it up for over 2 hours without showing any signs of weakness).
I'm actually not very worried about alkaline (or any) batteries leaking. There are also different kind of zinc-carbon batteries with a different chance of leaking. The cheapest ones do leak while the heavy duty type is often more leakproof than alkalines and dries out instead (I think that's why Varta called them "Super dry" at one time). These often come pre-installed in or packaged with remotes.
Most batteries that leaked for me did so in storage, and most of them were way past their expiry date. I also had batteries just break down without leaking although they were not actually used. This could actually explain the different charge state of batteries in one device like a remote... the weaker battery didn't get used more, but its chemistry broke down over the years.
I think the batteries most prone to such break down's are actually 9V's because they are built to deliver a relatively high current despite their small size, with the disadvantage being that they also self-discharge rapidly. I had a Duracell 9V that was barely used self-discharge in about 7 years. I got it when I played at the Christmas party of my company and my own 9V NiCd rechargable didn't last for the duration of the gig in the delay pedal I was using it in.
I do not try to avoid alkaline batteries because of leaking danger, but because of the economics. I did buy an 8-pack some years ago for the transmitter of my RC car because I figured I'd only rarely use it and it would probably last about 8 years at my rate of usage. And in 1999, I bought 10 Daimon alkaline "C" cells for a portable multitrack recorder for making some recordings for the public radio station I was member of back then. I only used them for about 4 field recordings until 2002, after that I stopped doing field recordings, and in 2005 I left the station. Although I didn't use them up, those 10 cells since have all broke down except for one which still powers my kitchen wall clock.
Other than that, I only use those alkaline or zinc-carbon batteries that come with devices. Sometimes, however, if they come with a high-drain device, I immediately replace them with rechargables and put the alkalines into storage (as I have done with my first electric toothbrush).
I never tried Lithium primaries, but I think those could be useful for those devices which don't work properly with the lower voltage of a NiMh battery because, contrary to Alkalines, they stay at nearly 1.5V for most of their life while Alkalines will suffer continuous voltage drop, so in those devices, Lithium primaries may actually achieve a much longer life than alkalines just because of this. However, I don't think I've got any such device in use currently.