I had a higher than expected OG in my homebrew. It was estimated to come out to 1.060 but instead ended up at 1.076. The back of the yeast I used said that it is good in a brew of up to 1.060 OG. The yeast seems to still be alive and kicking after 12 hours. Should I still be concerned? Is there anything I should do?
You don't need to be excessively concerned now. If the beer really was 1.076, the yeast will multiply to the quantity needed to ferment it - they don't just suddenly stop working when the OG goes above 1.060 - the yeast will still ferment and make beer. However, the multiplication is a stress factor for the yeast and the beer will not be as good as if you had pitched the correct amount of yeast. In future, buy more packs of yeast, or make a starter. This yeast calculator can tell you how much to pitch.
Worse case is that it may end up being slightly sweeter/fuller because of a higher final gravity, or possibly more fruity because of the increased rate of propagation, and maybe some hot alcohol as well from underpitching. But you'll still end up with beer.
To improve your process for future brews, I would try to find out why the beer was at 1.076. Assuming a beer kit with extract, the higher gravity will be either because you added less water than described or the wort wasn't completely mixed when you took a gravity sample, so some parts are thin while others are thick, and you took a reading from the thicker part. Or, if you formulated/made changes to a recipe then use brewing software such as Beersmith, or one of the many online recipe formulators.
It's a good idea to rehydrate dry yeast, especially for bigger beers, since you can lose up to 40% of the yeast cells when direct pitching. But of course, if there are instructions specifically saying to pitch directly then do that, otherwise rehydrate in the sachet in 200ml of hand-warm water and leave for 15 mins, then stir, and leave for a further 15 mins, before pitching.
For liquid yeast, always make a starter, since pitching just the pack/vial is almost always an underpitch. See Always making a starter vs following the package description.
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If you used liquid yeast, you did not pitch enough yeast. Despite what the yeast manufacturers say, a pack of liquid yeast alone should not be used for a beer over 1.040 OG. Both my experience and scientific pitching rate calculators confirm this. If it was dry yeast, you probably have just barely enough. No matter which kind you used, it would be a good idea to add another pack of the same type at this point.