How do I prevent or reduce shake when observing with binoculars?

For binoculars, your best option isn't just a simple tripod, but a parallelogram mount (link for representation only, not a product recommendation) on a tripod. While many standard to slightly larger than standard binoculars will have an adapter to attach them to a standard tripod; except for looking at things near the horizon it won't be comfortable to use. The problem is that because they're short the binocular eye pieces will often be too close to the tripod legs themselves for comfortable head positioning. If you're showing things to other people a parallelogram mount has the added bonus of being adjustable in height without changing what its pointed at.

If you're hand holding the most important thing you can do is to use both hands and brace your arms in some way. The top rail of a tall fence or a table (if you're seated) often work well. If you don't have anything else you can use your body itself by resting your upper arms against your chest to form a cantilever brace.

A last option, although not one I particularly recommend for other reasons (RSI injuries in particular) would be to get a job filling jars in an artisan scale canning/bottling plant. A friend of mine does that and after several years has bulked up his arms to the point he was able to use my 15x70 binoculars with 1 hand the way an average person could do with a light weight 5x35 pair.