I would change it, pronto
Two of my cars have the battery under covers and the other one is under the back seat. I never liked the short cables making it impossible to reverse the battery or the covers making it hard to look at the battery, but now I understand.
Iām wondering if this aftermarket 51R is simply too tall, or maybe the posts are a little too close, or too far apart, etc., and thatās why the hood prop is too close to the battery posts
Iāve run into this situation before, where one particular brand of battery just doesnāt fit well, even though it has the correct BCI number
This happened today to my 03 civic, hood prop became a massive conductor and yes it started a fire.
yikes. Sorry to hear that. 
My daughterās 03 Civic just went up in a flaming pile of smoke. Total loss. Change your battery to side terminals version or use a wooden pole insted. The arm was insolated but it was an exceptionally hot day yesterday and it melted through.
By going to side post, youāre opening up an entirely different can of worms . . .
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My Kia Forte has a thick plastic cover on the + terminal and the prop is hinged on the opposite side of the hood. The prop is also at least 12 inches short of reaching the battery. Itās refreshing to see actual preventive engineering.
My burnt car:
http://rgb.cc/03civic/side.jpg
Battery:
http://rgb.cc/03civic/battery.jpg
Hood arm self-welded to vehicle frame:
h ttp ://r gb.cc/03civic/con tact.jpg (Remove spaces, Iām only allowed to post two pictures)
This is obviously a manufacturer defect IMO. There is no way that this is safe and I am perplexed as to how their safety engineers signed off on it. I called Honda about it and they logged an incident report and told me that they have not heard of this happening before. Maybe Iām the first to call Honda about it.
If you have one of these and donāt want to risk injury or death, take a hack saw and cut that arm short where it doesnāt go anywhere near the battery. I wish I had.
Well, youāve proved it is possible to make a battery into a welder anyway, could come in handy in a pinch in event of a zombie attack ā¦ 
Seriously, you make an excellent point for all of us car owners to consider. On my Corolla the loose end of that arm is on the other side from the battery. On a car like yours it seems like a good idea to have a back-up system that prevents the arm from contacting the battery +, some sort of tough plastic shield or something. Itās configured similar to yours on my truck, so Iām going to take a look at it tomorrow in fact.
It seems that on most vehicles that I inspect the positive battery terminal cap is left dangling off to the side, people donāt have time to snap it into place and donāt believe anything could go wrong in the future.
@roberte1342 It is difficult to tell but by the location of the negative battery terminal clamp in the picture, was this the correct battery? The terminals should be at the rear of the battery. Group 51 or group 51R?
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Good question. The only time Iāve seen this happen is when the wrong battery is installed.
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And, Iād think youād have to neglect to snap the rod into its holder. However, it does appear to be āin harms wayā for the average motorist (not that observant):
I checked my Ford truck today, and I was wrong above, the loose end of the arm is away from the battery, the other side of the car, just like the Corolla. Good posts here for caution for those who have cars with the loose end of the hood prop rod closest to the battery. My Corolla has a plastic cap over the + post, but my truck has no plastic cap at all. I guess the auto-engineers didnāt come up with the plastic cap over the + post idea until after the early 70ās. Iām pretty sure my VW Rabbit had the plastic cap, late 1970s. The biggest threat for my truck having that problem is if the battery shifted, which is definitely possible the way it is configured. IMO batteries should be held down by a pair of sturdy nylon straps at 90 degree angles, like wrapping a Christmas present.
Iām not sure. I replaced it with an Interstate battery from Costco matching what was already in the car. The car had 1 prior owner. Iāll check my receipt. But even so I donāt think the positive and negative cables were long enough for the poles to be reversed.
OP: āI had a new battery installed (in my 2003 Honda Civic) by one of those āswift-oil-changeā places a few years agoā. Correct battery installed backwards? Of course the āmechanicā who noticed something was wrong should have just turned the battery around unless it was the same or another āIffy Lubeā.